TOWARDS A NON-LINEAR CALDER ´ ON-ZYGMUND THEORY GIUSEPPE MINGIONE To Tadeusz Iwaniec on his 60th birthday Contents 1. Road map 1 2. The linear world 2 3. Iwaniec opens the non-linear world 8 4. Review on measure data problems 20 5. Nonlinear Adams theorems 27 6. Beyond gradient integrability 36 References 47 1. Road map Calder´ on-Zygmund theory is a classical topic in the analysis of partial differential equations, and deals with determining, possibly in a sharp way, the integrability and differentiability properties of solutions to elliptic and parabolic equations, and especially of their highest order derivatives, once an initial, analogous information is known on the given datum involved. Now it happens that while a linear theory has been developed in a quite satisfactory way, a complete theory for gradient estimates for solutions to quasilinear equations of the type (1.1) div a(x, Du) = “suitable right hand side” is not yet developed, at least up to that complete extent one could wish for. The reason for such a difference lies of course in that linear structures allow, via certain explicit representation formulas, for applying rather abstract tools from Harmonic Analysis, semigroup theory, abstract Functional Analysis, interpolation theory and so forth. The use of all such tools is clearly ruled out in the case of non-linear structures as in (1.1), basically because representation formulas are not available, and more in general because no linear or sub-linear operator can be associated to the non-linear problem in question. The purpose of this paper is now to collect all those pieces - that is the avail- able theorems - that put together should form what we may call a non-linear Calder´ on-Zygmund theory. In fact we shall review some recent and less recent results concerning the integrability and (weak) differentiability properties of solu- tions to non-homogeneous equations involving operators of the type in (1.1), with a final emphasis on the content of a couple of recent papers we wrote [105, 106]. We would like to remark that a very deep, fully non-linear Calder´ on-Zygmund theory for fully non linear problems of the type (1.2) F (x, D 2 u)= f is available, being a fundamental contribution of Caffarelli - see [32, 31]. For obvi- ous reasons the phenomena and the techniques involved for the case (1.2), where 1
GIUSEPPE MINGIONE
Contents
1. Road map 1 2. The linear world 2 3. Iwaniec opens the non-linear
world 8 4. Review on measure data problems 20 5. Nonlinear Adams
theorems 27 6. Beyond gradient integrability 36 References 47
1. Road map
Calderon-Zygmund theory is a classical topic in the analysis of
partial differential equations, and deals with determining,
possibly in a sharp way, the integrability and differentiability
properties of solutions to elliptic and parabolic equations, and
especially of their highest order derivatives, once an initial,
analogous information is known on the given datum involved. Now it
happens that while a linear theory has been developed in a quite
satisfactory way, a complete theory for gradient estimates for
solutions to quasilinear equations of the type
(1.1) div a(x,Du) = “suitable right hand side”
is not yet developed, at least up to that complete extent one could
wish for. The reason for such a difference lies of course in that
linear structures allow,
via certain explicit representation formulas, for applying rather
abstract tools from Harmonic Analysis, semigroup theory, abstract
Functional Analysis, interpolation theory and so forth. The use of
all such tools is clearly ruled out in the case of non-linear
structures as in (1.1), basically because representation formulas
are not available, and more in general because no linear or
sub-linear operator can be associated to the non-linear problem in
question.
The purpose of this paper is now to collect all those pieces - that
is the avail- able theorems - that put together should form what we
may call a non-linear Calderon-Zygmund theory. In fact we shall
review some recent and less recent results concerning the
integrability and (weak) differentiability properties of solu-
tions to non-homogeneous equations involving operators of the type
in (1.1), with a final emphasis on the content of a couple of
recent papers we wrote [105, 106].
We would like to remark that a very deep, fully non-linear
Calderon-Zygmund theory for fully non linear problems of the
type
(1.2) F (x,D2u) = f
is available, being a fundamental contribution of Caffarelli - see
[32, 31]. For obvi- ous reasons the phenomena and the techniques
involved for the case (1.2), where
1
2 GIUSEPPE MINGIONE
solutions are intended in the viscosity sense and equations cannot
be differentiated, are quite different from the divergence
form/variational case (1.1), where a notion of weak solution in the
integral sense is adopted. For this reason we will not touch the
theory available for operators of the type (1.2), instead we refer
the reader to the excellent monograph [33]. On the other hand,
while the theory for (1.2) is essentially scalar, due to the fact
that a vectorial analog of viscosity solutions has not been
developed yet, we shall see that in the quasilinear case (1.1)
systems of PDE can be nevertheless dealt with, at least up to
certain extent.
A bridge between the viscosity methods and quasilinear structures
has been anyway built in [34], a paper who eventually inspired the
proof of many results for divergence form operators.
General notation. From now on, and for the rest of the paper, we
will use CZ as an acronym for Calderon-Zygmund. By we denote a
bounded open domain of Rn, with n ≥ 2. We shall denote by BR(x0)
the open ball in Rn of radius R and center x0, that is
BR(x0) = {x ∈ Rn : |x− x0| < R} . When the center will be
unimportant we shall simply denote BR(x0) ≡ BR. With BR ⊂ Rn being
a a ball with positive and finite radius, if g : BR → Rk is an
integrable map, the average of g over BR is
(g)BR := − ∫ BR
g(x) dx .
When considering a function space X(,Rk) of possibly vector valued
measurable maps defined on an open set ⊂ Rn, with k ∈ N, e.g.:
Lp(,Rk),W β,p(,Rk), we shall define in a canonic way the local
variant Xloc(,Rk) as that space of maps f : → Rk such that f ∈
X(′,Rk), for every ′ ⊂⊂ . Moreover, also in the case f is vector
valued, that is k > 1, we shall also use the short hand notation
X(,Rk) ≡ X(), or even X(,Rk) ≡ X when the domain is not important,
and we want to emphasize a qualitative property.
Finally, several times, in order to simplify the exposition, we
shall not specify the domain of integration when stating certain
results; in such cases we shall mean that the domain is not
important, or that the result in question holds in a local way, and
then also up to the boundary provided suitable boundary conditions
are made. Other times, the domain considered will be simply the
whole space Rn.
Acknowledgments. This research is supported by the ERC grant 207573
“Vec- torial Problems”.
2. The linear world
The material in this section is classical, and we are just giving a
short survey of results in order to settle a background of linear
results to later present in a more efficient way the forthcoming
non-linear ones. We shall prefer here a more informal presentation,
not giving all the details but rather aiming to give a general
overview. References for the next results and to more details can
be for instance [53, 64, 116].
2.1. Lebesgue spaces. The basic example is at this stage the
Poisson equation
(2.1) 4u = f ,
which for simplicity we shall initially consider in the whole Rn,
for n ≥ 2; here f is the datum. The previous equation, as all the
other ones considered in this section is satisfied in the
distributional sense, while all solutions are supposed to be at
least of class W 1,1.
TOWARDS A NON-LINEAR CALDERON-ZYGMUND THEORY 3
A classical result, going back to the fundamental work of Calderon
& Zygmund asserts the solvability in the right function
spaces
(2.2) f ∈ Lγ =⇒ D2u ∈ Lγ for every γ > 1 .
Of course the last result comes along together with an a priori
estimate
(2.3) D2uLγ . fLγ for every γ > 1 .
We remark that well-known counterexamples show that the same
implication is false for γ = 1, even locally. As a consequence of
(2.2) and of Sobolev embedding theorem we have also
(2.4) f ∈ Lγ =⇒ Du ∈ L nγ n−γ for every γ ∈ (1, n) .
The classical proof of (2.2)-(2.4) goes via a representation
formula involving the so called fundamental solution, that is the
Green’s function, say for n ≥ 2
(2.5) u(x) ≈ ∫ G(x, y)f(y) dy
where
log |x− y| if n = 2 .
Then, after differentiating twice (2.5) one arrives at a new
representation formula
(2.7) D2u(x) ≈ ∫ K(x− y)f(y) dy
where now K(·, ·) is a so called CZ kernel, that is
(2.8) KL∞ ≤ B ,
where K denotes the Fourier transform ofK(·), and moreover the
following Hormander cancelation condition holds:
(2.9)
|K(x− y)−K(x)| dx ≤ B for every y ∈ Rn .
A this point the standard CZ theory of singular integrals comes
into the play: the linear operator
f 7→ I0(f)
where, in fact
∫ K(x− y)f(y) dy ,
is bounded from Lγ to Lγ , for every γ ∈ (1,∞) and therefore (2.2)
follows from (2.7). Related a priori estimates for solutions to
(2.1) follow from the a priori bounds on I0 in the various function
spaces involved. The crucial point in the Calderon-Zygmund theory
of singular integrals is that although the kernel K(·) is singular
- i.e. not integrable - in the sense that
|K(x)| . 1
|x|n ,
condition (2.9) encodes enough cancelations to ensure the
convergence of I0(f) in Lγ for γ > 1.
Related to the equation (2.1) is the following one:
(2.11) 4u = div F ,
div Du = div F ,
4 GIUSEPPE MINGIONE
we expect, for homogeneity reasons - that is Du scales as F - that
Du enjoys the same integrability of F . This is the case, indeed it
holds that
(2.12) DuLγ . FLγ for every γ > 1 ,
as desired, and this is again achievable via the use of singular
integrals [68, Pro- porition 1].
As mentioned above, (2.4) follows from (2.2) applying Sobolev
embedding the- orem, but there is another, more direct way to get
it, without appealing to the the theory of singular integrals, but
rather relying on a lighter one: that of Riesz potentials, also
called fractional integrals. In fact, once again starting from
(2.5), but differentiating it once we gain yet another
representation formula
(2.13) Du(x) ≈ ∫ K1(x, y)f(y) dx ,
where, accordingly
This motivates the introduction of so called fractional
integrals.
Definition 2.1. Let β ∈ [0, n); the linear operator, acting on
measurable functions and defined by
Iβ(f)(x) :=
∫ Rn
f(y)
is called the β-Riesz potential of f .
Needless to say it is possible to define the action of Riesz
potentials over measures with finite total mass as follows as
Iβ(µ)(x) :=
∫ Rn
dµ(y)
As a matter of fact the following theorem holds:
Theorem 2.1 ([66]). Let β ∈ [0, n); for every γ > 1 such that βγ
< n we have
(2.15) Iβ(f) L
≤ c(n, β, γ)fLγ(Rn) .
See also [99, Theorem 1.33] for a proof. At this point the
derivation of (2.4) is straightforward from (2.13), (2.14) and
(2.15). This is not a surprise since a less general version of
Theorem 2.1 was used - and actually re-derived - by Sobolev in
order to prove his celebrated embedding theorem for the case t >
1 - the one for the case t = 1 actually necessitates other tools,
that is the celebrated Gagliardo- Nirenberg inequalities.
Remark 2.1. There is a basic difference between fractional and
singular integrals: in the theory of fractional integrals one does
not use cancelation properties of the kernel as the one in (2.9).
Indeed estimate (2.15) only uses the size of |x − y|β−n, and the
constant c(n, β, γ) blows-up for β → 0, if using the technique
leading to Theorem 2.1. In other words the crucial fact is that for
β > 0 the function
1
TOWARDS A NON-LINEAR CALDERON-ZYGMUND THEORY 5
2.2. The borderline case γ = 1. A natural question raises now: What
happens in the borderline case γ = 1? The results presented up to
now fail, but an answer can be nevertheless obtained by considering
suitable functions spaces. Let me start by the so called
Marcinkiewic spaces Mt(A,Rk), also called Lorentz spaces and
denoted by Lt,∞(A), or by Ltw(A), when they are called “weak-Lt”
spaces, or Lorentz spaces - see Definition 5.4 below.
Definition 2.2. Let t ≥ 1. A measurable map w : → Rk belongs
toMt(,Rk) ≡ Mt() iff
(2.16) sup λ≥0
λt|{x ∈ A : |w| > λ}| =: wtMt() <∞ .
It turns out that linear CZ integral operators send L1 intoM1;
therefore, in the borderline case γ = 1 estimates (2.3) and (2.12)
turn to
(2.17) D2uM1 . fL1 ,
and
(2.19) Iβ(f) M
so that for the Poisson equation (2.1) it holds that
(2.20) f ∈ L1 =⇒ Du ∈M n n−1 .
For the limiting embedding property of Riesz potential see the
classical paper of Adams [6]; this paper contains results we shall
examine in greater detail later.
Remark 2.2. There is a problem in (2.17)-(2.18); these have to be
thought as a priori estimates, since the derivatives involved there
are not the distributional ones. To avoid complications, we shall
consider (2.17)-(2.18) when f , F are smooth, and therefore u is
also smooth, and we shall retain (2.17)-(2.18) only in such a
qualitative form.
The importance of the spaceMt clearly lies in the fact that it
serves to describe in a sharp way certain limiting integrability
situations, very often occurring in modern non-linear analysis, as
those given by potential functions. In fact, the prototype of Mt
functions is given by the potential |x|−n/t; note that
(2.21) 1
In general the following inclusions hold:
(2.22) Lt $Mt & Lt−ε for every ε > 0 .
As for the first one, observe that
|{|w| > λ}| =
∫ {|w|>λ}
so that wMt ≤ wLt
holds, and in fact the estimation in (2.23) motivates the
definition of Marcinkiewicz spaces. Marcinkiewicz spaces are
nowadays of crucial importance in the analysis of
6 GIUSEPPE MINGIONE
problems with critical non-linearities as those involving harmonic,
p-harmonic, and bi-harmonic maps, Euler equations and other pdes
from fluid-dynamics.
Inclusions (2.22) tell us that Marcinkiewicz spaces interpolate
Lebesgue spaces; we shall see later a more refined way of
interpolating Lebesgue spaces, when Lorentz spaces will be
introduced, extending both Marcinkiewicz and Lebesgue spaces, and
introducing finer scales for measuring the size of a function. For
more on Marcinkiewicz spaces we refer to Section 5.3 below.
The second natural question is now to find a condition for which in
estimates (2.17)-(2.18) we can obtain the full Lebesgue
integrability scale instead of the Marcinkiewicz one. The question
is the same that finding a function space “slightly smaller” than
L1, which is mapped into L1 by singular integral operators. For the
developments we are concerned with the answer is actually
doublefold.
In order to replace M1 by L1 in the left hand side of (2.17) one
has to consider a slightly larger space for the right hand datum f
, called Hardy space and denoted by H1. The elements of such space
are functions enjoying enough cancelation prop- erties to match
with those by singular integrals and finally yielding convergence
in L1. The story of the Hardy spaces is essentially a complex
function theory one until the fifties, when a completely real
function theory characterization of such spaces was settle down
mainly by authors like Stein and Weiss; see the classical paper of
Fefferman & Stein [58]. A central concept is the one of atomic
decompo- sition, initially due to Coifman [38], see also [92],
allowing to give a particularly simple definition which we adopt
here. We recall that an atom a over a cube Q is a function such
that supp a(·) ⊆ Q and moreover∫
Q
|Q| ,
hold. The atom a(·) can actually be thought as a bump function
exhibiting cance- lations. Then a function f belongs to the Hardy
H1(Rn) iff there exists a sequence of atoms {ak} such that the
following atomic decomposition holds:
(2.24) w(x) = ∑ k∈N
λkak(x) and ∑ k∈N |λk| <∞ .
The inf of the sums ∑ |λk| over all possible such decompositions
naturally defines
the Hardy space norm of w. We won’t any longer deal with Hardy
spaces here; this in another story, too much unrelated to the
non-linear setting we are going to switch to in the next sections.
As a matter of fact the atomic decompositions of Hardy functions
allows a perfect match with the cancelations properties of the CZ
kernel. In fact, using both the cancelation properties of CZ
kernels and the zero-average property of a over the cube Q, it is
easy to see that I0(a)L1 ≤ c, where the constant c ultimately
depends on the constant B occurring in (2.8)-(2.9), but is actually
independent of the atom considered a. Using this fact, and the very
definition of Hardy spaces, that is the decomposition in (2.24),
the boundedness in H1 follows:
I0(f)L1 . fH1 .
Such a proof cannot be apparently extended to the case of
non-linear operators, in that such a kind of delicate cancelation
phenomena are apparently destroyed by general non-linearities, in
other words, non-linear operators seems not to read cancelation
properties of the Hardy functions.
It remains to try with additional size conditions. The space L
logL(), with ⊆ Rn being a bounded domain, is therefore defined as
those of the functions f satisfying ∫
TOWARDS A NON-LINEAR CALDERON-ZYGMUND THEORY 7
Such space, a particularly important instance of what are called
Orlicz spaces, becomes a Banach space when equipped with the
following Luxemburg norm:
(2.25) wL logL() := inf
{ λ > 0 : −
} <∞ .
Note that for homogeneity reasons that will be clear later we have
incorporated in the above definition a dependence on the measure ,
by considering an averaged integral in (2.25). The following
equivalence due to T. Iwaniec [73]:
(2.26) wL logL() ≈ − ∫
) dx =: |w|L logL() ,
and the striking fact is that the last quantity actually defines a
true norm in L logL(), which is therefore equivalent to the usual
Luxemburg one (2.25) via a constant independent of the domain . An
obvious consequence of the defini- tions above is the following
inclusion:
L logL() $ L1() and − ∫
|w| dx ≤ wL logL() .
As a matter of fact the space L logL is sent into L1 by singular
integrals operators
I0(f)L1 . fL logL .
as a consequence we have limiting L1-estimates in (2.3) and (2.12):
these turn to
(2.27) D2uL1 . fL logL ,
and
respectively, while (2.15) turns to
(2.29) Iβ(f) L
so that for the Poisson equation (2.1) it holds that
(2.30) f ∈ L logL =⇒ Du ∈ L n n−1 .
For the last two results we again refer to Adams’ paper [6].
2.3. Perturbations of the linear theory. The results on the
differentiability properties explained in Section 2.1 extend to
linear elliptic equations of the type
(2.31) AD2u = f
ν|λ|2 ≤ Aλ, λ ≤ L|λ|2
holds for any λ ∈ Rn. In fact, up to changing coordinates, the
equation (2.31) behaves as the usual Poisson equation. The same
results hold for equations with
variable, continuous coefficients, that is A ≡ A(x) ∈
C0(Rn,Rn2
); in fact in this case, due to the continuity of the coefficients,
the equation can be considered as a local perturbation of the
Laplace operator, and CZ estimates follow by mean of local
perturbation and fixed point arguments; see for instance [62,
Paragraph 10.4].
It is now obvious that the same results cannot hold when the
coefficient matrix A(x) is just measurable, due to well-known
counterexamples. Anyway certain types of mild discontinuities for
the matrix A(·) can still be allowed. This is the case when the
entries of A(x) are assumed to be VMO functions - see Section 3
below. These are functions which are basically continuous up to an
averaging process.
The first results in this direction have been obtained by Chiarenza
& Frasca & Longo [39] by using a few deep theorems from
Harmonic Analysis on the bounded- ness of commutator operators
involving BMO coefficients. This strategy, applied
8 GIUSEPPE MINGIONE
to a variety of problems with different boundary conditions,
heavily relies on the linearity of the problem considered, in that
it is based on the use of representation formulas via fundamental
solutions. This approach, relying on a series of sophisti- cated
tools, can be actually completely bypassed, via the use of suitable
maximal function operators, as we shall see more in detail in
Section 3.3 below; moreover various type of boundary value problems
with very weak regularity assumptions on the boundary are
treatable, see for instance [28] and related references.
2.4. “Extremals” of the linear theory. In the last years the
problem of deter- mining a CZ for linear problem not involving
treatable kernels - for instance kernels which are not regular
enough to satisfy Hormander condition (2.9) - has been very often
dealt with. This are problems in which the linearity of the
equations con- sidered still allows to apply - in suitably tailored
sophisticated forms - abstract methods from operator theory,
interpolation theory, and Harmonic Analysis. Very often, for
instance, the linearity of the equations involved suffices to
associate to them a suitable sub-linear operator, so that
interpolation methods and techniques are applicable. For such
developments we refer the reader to the interesting papers of
Auscher & Martell [13, 14, 15], and their related
references.
Plan for the next sections. At the end of this introductory part we
would like to conclude with a few remarks, and a plan for the next
sections. The very basic survey of results in this section settles
the mood for the rest of the paper. The purpose here is to describe
some extension of the integrability properties in (2.2) and (2.4)
to solutions to non-linear elliptic problems. We would like to
emphasize the doublefold character of the issue: basic
integrability of the gradient in (2.4), and second order
differentiability in (2.2), which is the maximal regularity result
since equation (2.1) is a second order one. Of course, in general
(2.2) implies (2.4) via Sobolev embedding theorem, but there are
anyway cases in which (2.4) holds, but (2.2) does not, unless using
additional assumptions. The case of equations with low regularity
coefficients is an instance. Section 3 will be dedicated to results
concerning the integrability of the gradient for large exponents,
and we shall deal with classical weak or energy solutions. Sections
4 and 5 will be still dedicated to integrability issues, this time
for small exponents. We shale therefore deal with so-called very
weak solutions, i.e. solutions not belonging to the natural energy
space. In particular, in Section 4 we shall give a rapid
introduction to measure data problems, including basic regularity
results. In Section 5 we shall present more delicate integrability
results in different scales of spaces as Morrey spaces, and in
finer scales as Lorentz spaces. Finally in Section 6 we shall
concentrate on the higher differentiability of solutions.
3. Iwaniec opens the non-linear world
3.1. The notion of solution. The general setting we are going to
examine con- cerns non-linear equations and systems which in the
most general form look like
(3.1) −div a(x, u,Du) = H in ,
where a : ×RN×RNn → RNn is a measurable vector field, with n ≥ 2
and N ≥ 1, which is continuous in the last two arguments, and
initially satisfies the following p-growth assumption:
(3.2) |a(x, u, z)| ≤ L(1 + |z|2) p−1 2 for p > 1 .
When N = 1 (3.1) reduces to an equation and we are in the scalar
case. On the right hand side of (3.1) we initially assume
that
H ∈ D′(,RN ) .
TOWARDS A NON-LINEAR CALDERON-ZYGMUND THEORY 9
The notion of distributional solution prescribes that a map u ∈ W
1,1(,RN ) is a weak solution to (3.1) iff u is such that a(x, u,Du)
∈ L1(,RN ) and satisfies
(3.3)
∫
a(x, u,Du)Ddx = H, for every ∈ C∞c (,RN ) .
As it will be clear later, this definition is too general.
Therefore we recall the following:
Definition 3.1. An energy solution to (3.1), under the assumption
(3.2), is a distributional solution in the sense of (3.3) enjoining
the additional property
u ∈W 1,p 0 (,RN ) .
In turn, when dealing with energy solutions, the natural conditions
to impose on H is of course H ∈ W−1,p′(); this allow to test (3.3)
with W 1,p
0 () functions - this follows by simple density arguments - and in
particular with multiples of the solution itself. In turn, when
having H ∈ W−1,p′() and a suitable mono- tonicity assumption on
a(·), the existence of an energy solution follows by classical
monotonicity methods. For this we refer to the classical
[98].
In the rest of this section, unless otherwise stated, we shall deal
with the notion of energy solution; non-energy solutions will
appear later, when dealing with measure data problems.
3.2. A starting point. The starting point here is the following
natural p-Laplacean analog of equation (2.11):
(3.4) div (|Du|p−2Du) = div (|F |p−2F ) for p > 1 ,
which indeed reduces to (2.11) for p = 2. Note that the right hand
side of (3.4) is written in the peculiar form div (|F |p−2F ) in
order to facilitate a more elegant presentation of the results, and
also because such form naturally arises in the study of certain
projections problems motivated by multi-dimensional quasi-conformal
geometry [68]. Anyway, one could immediately consider a right hand
side of the type div G by on obvious change of the vector
field
G ≡ |F |p−1 F
|F | ⇐⇒ F ≡ |G|
1 p−1
G
|G| .
The following fundamental result in essentially due to Tadeusz
Iwaniec, who, in the paper [68] established the foundations of the
non-linear CZ theory.
Theorem 3.1 ([68]). Let u ∈ W 1,p(Rn) be a weak solution to the
equation (3.4) in Rn. Then
F ∈ Lγ(Rn,Rn) =⇒ Du ∈ Lγ(Rn,Rn) for every γ ≥ p .
The local version of this result is
Theorem 3.2. Let u ∈ W 1,p() be a weak solution to the equation
(3.4) in , where is a bounded domain in Rn. Then
(3.5) F ∈ Lγloc(,Rn) =⇒ Du ∈ Lγloc(,Rn) for every γ ≥ p .
Moreover, there exists a constant c ≡ c(n, p, γ) such that for
every ball BR b it holds that
(3.6)
( − ∫ BR/2
+ c
( − ∫ BR
.
10 GIUSEPPE MINGIONE
A proof can be adapted from [3], for instance. From now on, for
ease of presen- tation we shall confine ourselves to treat local
regularity results.
The non-trivial extension to the case when (3.4) is a system has
been obtained by DiBenedetto & Manfredi, who caught a
borderline case too.
Theorem 3.3 ([49]). Let u ∈W 1,p(,RN ) be a weak solution to the
system (3.4), where is a bounded domain in Rn, and N ≥ 1. Then
(3.5) holds. Moreover
(3.7) F ∈ BMOloc(,RNn) =⇒ Du ∈ BMOloc(,RNn) .
3.3. BMO and VMO functions. There appears a new function space in
(3.7), the space of functions with bounded mean oscillations,
introduced by John & Niren- berg [74]. In order to introduce
BMO functions let us introduce the quantity
(3.8) [w]R0 ≡ [w]R0, := sup
BR⊂,R≤R0
|w(x)− (w)BR | dx .
Then a measurable map w belongs to BMO() iff [w]R0 < ∞, for
every R0 < ∞. It turns out that BMO ⊂ Lγ for every γ <∞,
while a deep and celebrated result of John & Nirenberg tells
that every BMO function actually belongs to a suitable weak Orlicz
space generated by an N-function with exponential growth [74], and
depending on the BMO norm of w. Specifically, we have
(3.9) |{x ∈ QR : |w(x)− (w)QR | > λ}| ≤ c1(n) exp
( − c2λ
[w]2R,QR
) where QR is a cube whose sidelength equals R, and c1, c2 are
absolute constants. Anyway BMO functions can be unbounded, as shown
by log(1/|x|). For the proof of (3.9) a good reference is for
instance [53, Theorem 6.11].
Related to BMO functions are functions with vanishing means
oscillations. These have been originally defined by Sarason [110]
as those BMO functions w, such that
lim R→0
[w]R, = 0 .
In this way one prescribes a way to allow only mild
discontinuities, since the oscil- lations of w are measured in an
integral, averaged way.
As outlined in Section 2.3, the linear CZ theory can be extended to
those prob- lems/operators involving VMO coefficients. This happens
also in the non-linear case, as proved by Kinnunen & Zhou who
considered a class of degenerate equa- tions whose model is given
by
(3.10) div (c(x)|Du|p−2Du) = div (|F |p−2F ) for p > 1 ,
where the coefficient c(·) is a VMO function satisfying
(3.11) c(·) ∈ VMO() and 0 < ν ≤ c(x) ≤ L <∞ .
The outcome is now
Theorem 3.4 ([84]). Let u ∈W 1,p() be a weak solution to the
equation (3.10) in , where is a bounded domain in Rn, and the
function c(·) satisfies (3.11). Then assertions (3.5)-(3.6) hold
for u, and the constant in estimate (3.6) also depends on the
coefficient function c(·).
Before going on, we will comment on the strategy adopted in order
to prove Theorems 3.1-3.4; in turn, up to non-trivial complications
due to the increasing level of generality, this goes back to the
original Iwaniec’s paper [70]. The idea of Iwaniec is very natural;
in the linear case the estimates are obtained using exactly two
ingredients: global representation formulas as in (2.5), and then
the use of CZ
TOWARDS A NON-LINEAR CALDERON-ZYGMUND THEORY 11
theory of singular integrals (2.10). Then Iwaniec replaces the
first ingredient using a local comparison argument with solutions
to the homogeneous p-Laplace equation
(3.12) div (|Dv|p−2Dv) = 0 .
The pointwise regularity estimates of Uhlenbeck-Ural’tseva [121,
122] are then ap- plied in order to provide an analog of the local
representation formula in the linear case. Then finally Iwaniec
shows that it is possible to pointwise estimate the sharp maximal
function of Du with the maximal function of the datum F , and the
con- clusion follows by applying the well-known maximal theorems of
Hardy-Littlewood, and Fefferman-Stein, which, at this stage, play
the role of the boundedness of sin- gular integrals.
3.4. More general operators. We will now turn to more general
equations of the type
(3.13) div a(x,Du) = div (|F |p−2F ) for p > 1 ,
where a : × Rn → Rn is a continuous vector field satisfying the
following strong p-monotonicity and growth assumptions:
(3.14)
1 2 |Da(x, z)| ≤ L(s2 + |z|2)
p−1 2
ν(s2 + |z|2) p−2 2 |λ|2 ≤ Da(x, z)λ, λ
|a(x, z)− a(x0, z)| ≤ Lω (|x− x0|) (s2 + |z|2) p−1 2 ,
whenever x, x0 ∈ , z, λ ∈ Rn. Here we take p > 1, s ∈ [0, 1],
while ω(·) is a modulus of continuity, that is a non-decreasing
function defined on [0,∞] such that
(3.15) lim t→0+
ω(t) = 0 .
We have seen from the previous section that the regularity of
solutions to homoge- neous related homogeneous equations
(3.16) div a(x,Dv) = 0 ,
is an important ingredient in the proof of the gradient estimates,
in that the regu- larity estimates for solutions to (3.16) are then
used in a comparison scheme to get proper size estimates for the
gradient of solutions to (3.13). Therefore in order to state a
theorem of the type 3.2 one has to consider operators such that
solutions v to (3.16) enjoy the maximal regularity, which in our
case is Dv ∈ Lγ for every γ <∞. On the other hand, an obvious a
posteriori arguments is that if an analog of Theorem 3.2 would hold
for equation (3.13), then applying it with the choice F ≡ 0 would
in fact yield Dv ∈ Lγ for every γ <∞.
This is the case for solutions to (3.16) under assumptions (3.14).
Therefore it holds the following:
Theorem 3.5. Let u ∈ W 1,p() be a weak solution to the equation
(3.13), where is a bounded domain in Rn, and such that assumptions
(3.14) are satisfied. Then (3.5) holds for u and moreover
(3.17)
( − ∫ BR/2
,
where c ≡ c(n, p, ν, L, γ).
See for instance [3], from which a proof of the previous result can
be adapted. There is a number of possible variants of the previous
result; we shall outline a couple of them.
12 GIUSEPPE MINGIONE
The first deals with non-linear operators with VMO coefficients,
more precisely with equations of the type
(3.18) div [c(x)a(Du)] = div (|F |p−2F ) for p > 1 ,
where the vector field a : Rn → Rn satisfies (3.14) - obviously
recast for the case where there is no x-dependence, while the
coefficient function c(·) satisfies (3.11). We have
Theorem 3.6. Let u ∈ W 1,p() be a weak solution to the equation
(3.18) in , where is a bounded domain in Rn, and such that
assumptions (3.11) and (3.14) are satisfied. Then the assertion in
(3.5) and (3.17) hold for u, and the constant c appearing in (3.17)
depends also on the coefficient c(·).
For a proof one could for instance adapt the arguments from [3,
107]. In partic- ular, in the last paper a version of Theorem 3.6
in the so called Heisenberg group has been obtained, while in the
first one CZ estimates have been obtained for a class of
non-uniformly elliptic operators. The previous result can be also
extended to the boundary when considering the Dirichlet problem -
as (3.23) below, under very mild assumptions on the regularity of
the boundary ∂; we will not deal very much with boundary
regularity, and for such issues we for instance refer to [28] and
related references.
The second generalization goes in another direction; we have seen
that the pos- sibility of getting a priori regularity estimates for
homogeneous equations as (3.16) is crucial for proving related CZ
estimates. As matter of fact the assumptions guaranteeing that
solutions to
(3.19) div a(Dv) = 0 ,
are Lipschitz can be considerably relaxed with respect to those in
(3.14). More precisely we may consider a : Rn → Rn to be a
continuous vector field satisfying
(3.20)
ν(s2 + |z1|2 + |z2|2) p−2 2 |z2 − z1|2 ≤ a(z2)− a(z1), z2 −
z1
|a(z)| ≤ L(s2 + |z|2) p−1 2 ,
whenever z1, z2 ∈ Rn, where p > 1 and s ∈ [0, 1]. Note that here
the vector field a(·) is not even assumed to be differentiable. We
then have
Theorem 3.7. Let u ∈W 1,p() be a weak solution to the
equation
div a(Du) = div (|F |p−2F ) ,
where is a bounded domain in Rn, and such that assumptions (3.20)
are satisfied. Then (3.5) and (3.17) hold for u.
The proof can be obtained using the method in [3], but using the a
priori reg- ularity estimates for solutions to (3.19) developed for
instance [85, 57] - note that the proofs there extend to the
general non-variational case of general operators in divergence
form. Finally we remark that in Theorem 3.7 we can allow VMO
coefficients as in (3.18).
3.5. The case of systems. Theorem 3.3 tells us that CZ estimates
extend to the case of systems when considering the specific
p-Laplacean system. We now wonder up which extend the results of
the previous section extend to general systems. The reason for
Theorem 3.3 to hold is that, as first shown by Uhlenbeck, solutions
to the homogeneous p-Laplacean system (3.12) are actually of class
C1,α for some α > 0. This makes the local comparison argument
work, finally leading to Theorem 3.3. In fact a major effort in
[49] is to show suitable form of a priori estimates for solutions
to (3.12). We also recall that, as pointed out in the previous
section, the regularity
TOWARDS A NON-LINEAR CALDERON-ZYGMUND THEORY 13
of solutions to associated homogeneous problems is crucial to
obtain the desired CZ estimates.
In the case of general systems as (3.13), and satisfying (3.14), we
cannot expect a theorem like 3.3 to hold, and for a very simple
reason. It is known that solutions to general homogeneous systems
as
(3.21) div a(Dv) = 0 ,
are not everywhere regular; they are C1,α-regular only when
considered outside a closed negligible subset of , in fact called
the singular set of the solution. Moreover, even for p = 2, and in
the case of a smooth vector field a(·), Sverak & Yan [113] have
shown that solutions to (3.21) may even be unbounded in the
interior of ; for such issues see for instance the recent survey
paper [103]. This rules out the validity of Theorem 3.3 for general
systems in that, should it hold, when applied to the case (3.21) it
would imply the everywhere Holder continuity of v in , clearly
contradicting the existence of unbounded solutions proved in
[113].
On the other hand an intermediate version of Theorem 3.3 which is
valid for general systems holds in the following form:
Theorem 3.8 ([86]). Let u ∈W 1,p(,RN ) be a weak solution to the
system
div a(x,Du) = div (|F |p−2F ) ,
for N ≥ 1, where is a bounded domain in Rn and the continuous
vector field a : × RNn → RNn satisfies (3.14) when suitably recast
for the vectorial case. Then there exists δ ≡ δ(n,N, p, L/ν) > 0
such that
F ∈ Lγloc(,RNn) =⇒ Du ∈ Lγloc(,RNn) ,
n− 2 + δ when n > 2 ,
while no upper bound is prescribed on γ in the two-dimensional case
n = 2. More- over, the local estimate (3.17) holds.
Note that the previous theorem does not contradict the
counterexample in [113], since this does not apply when n = 2. The
previous result comes along with a global one. For this we shall
consider the Dirichlet problem
(3.23)
u = v on ∂
for some boundary datum v ∈ W 1,p(,RN ); here we assume for
simplicity that ∂ ∈ C1,α, but such an assumption can be relaxed.
The main result for (3.23) is
Theorem 3.9 ([86]). Let u ∈W 1,p(,RN ) be the solution to the
Dirichlet problem (3.23) for N ≥ 1, where is a bounded domain in Rn
and the continuous vector field a : ×RNn → RNn satisfies (3.14)
when suitably recast for the vectorial case. Then there exists δ ≡
δ(n,N, p, L/ν) > 0 such that∫
(|Dv|γ + sγ) dx ,
holds whenever (3.22) is satisfied for n > 2, while no upper
bound is imposed on γ in the two-dimensional case n = 2; the
constant c depends only on n,N, p, ν, L, γ, ∂.
The previous theorem reveals to be crucial when deriving certain
improved bounds for the Hausdorff dimension of the singular set of
minima of integral func- tions - see [86] - and when proving the
existence of regular boundary points for solu- tions to Dirichlet
problems involving non-linear elliptic systems - see [54]. Moreover
the peculiar upper bound on γ appearing in (3.22) perfectly fits
with the parameters
14 GIUSEPPE MINGIONE
values in order to allow the convergence of certain technical
iterations occurring in [86, 54].
The proof of Theorems 3.8-3.9 is based on an argument different
from those in [70], but rather relying on some more recent methods
used by Caffarelli & Peral [34] in order to prove higher
integrability of solutions to some homogenization problems.
Although quite different from the previous ones, such method still
relies on the use of maximal operators.
3.6. Parabolic problems. The extension to the parabolic case of the
results of the previous sections is quite non-trivial, and in fact
the validity of Theorem 3.2 for the parabolic p-Laplacean
system
(3.24) ut − div (|Du|p−2Du) = div (|F |p−2F )
remained an open problem for a while in the case p 6= 2, even in
the case of one scalar equation N = 1; it was settled only recently
in [4]. All the parabolic problems in this section, starting by
(3.24), will be considered in the cylindrical domain
(3.25) T := × (0, T ) ,
where, as usual, is a bounded domain in Rn, and T > 0. Let us
now explain where are the additional difficulties coming from. As
we
repeatedly pointed out in the previous sections, the proof of the
higher integrability results strongly relies on the use of maximal
operators. This approach is completely rules out in the case of
(3.24). This is deeply linked to the fact that the homogeneous
system
(3.26) ut − div (|Du|p−2Du) = 0
locally follows an intrinsic geometry dictated by the solution
itself. This is essentially DiBenedetto’s approach to the
regularity of parabolic problems [46] we are going to briefly
streamline - see also Remark 3.1. The right cylinders on which the
problem (3.26) enjoys good a priori estimates when p ≥ 2 are of the
type
(3.27) Qz0(λ2−pR2, R) ≡ BR(x0)× (t0 − λ2−pR2, t0 + λ2−pR2) ,
where z0 ≡ (x0, t0) ∈ Rn+1 and the main point is that λ must be
such that
(3.28)
∫ Qz0 (λ2−pR2,R)
|Du|p ≈ λp .
The last line says that Qz0(λ2−pR2, R) is defined in an intrinsic
way. It is actually the main core of DiBenedetto’s ideas to show
that such cylinders can be constructed and used. Now the point is
very simple: since the cylinders in (3.27) depend on the size of
the solution itself, then it is not possible to associate to them,
and therefore to the problem (3.26), a universal family of
cylinders - that is independent of the solution considered. In turn
this rules out the possibility of using parabolic type maximal
operators.
In the paper [4] we overcame this point by introducing a completely
new tech- nique bypassing the use of maximal operators, and giving
the first Harmonic Anal- ysis free, purely pde proof, of non-linear
CZ estimates. The result is split in the case p ≥ 2 and p <
2.
Theorem 3.10 ([4]). Let u ∈ C(0, T, L2(,RN ))∩Lp(0, T,W 1,p(,RN ))
be a weak solution to the parabolic system (3.24), where is a
bounded domain in Rn, and p ≥ 2. Then
(3.29) F ∈ Lγloc(T ,RNn) =⇒ Du ∈ Lγloc(T ,RNn) for every γ ≥ p
.
TOWARDS A NON-LINEAR CALDERON-ZYGMUND THEORY 15
Moreover, there exists a constant c ≡ c(n,N, p, ν, L, γ) such that
for every parabolic cylinder QR ≡ BR(x0)× (t0 −R2, t0 +R2) b T it
holds that(
− ∫ QR/2
) 1 p
] p 2
.(3.30)
We note the peculiar form of the a priori estimate (3.30), which
fails to be a reverse Holder type inequality as (3.17) due to the
presence of the exponent p/2, which is the the scaling deficit of
the system (3.24). The presence of such exponent is natural, and
can be explained as follows: in fact, let us consider the case F ≡
0, that is (3.26). We note that if u is a solution, then, with c ∈
R being a fixed constant, the function cu fails to be a solution of
a similar system, unless p = 2. Therefore, we cannot expect
homogeneous a priori estimate of the type (3.17) to hold for
solutions to (3.24), unless p = 2, when (3.30) becomes in fact
homogeneous. Instead, the appearance of the scaling deficit
exponent p/2 in (3.30) precisely reflects the lack of homogeneity.
Another sign of the lack of scaling is the presence of the additive
constant in the second integral, this is a purely parabolic fact,
linked to the presence of a diffusive term - that is ut - in the
system.
Remark 3.1 (Intrinsic geometry and self-rebalancing). We will
explain here some basic principles of DiBenedetto’s intrinsic
geometry [46], confining ourselves to the case p > 2; the case p
< 2 can be treated by similar means. The reason for consider-
ing cylinders as in (3.27) appears natural if we use the following
heuristic argument: the relation (3.28) roughly tells us that |Du|
≈ λ in the cylinder Qz0(λ2−pR2, R). Therefore in the same cylinder
we may think to system (3.26) as actually
(3.31) ut − div (λp−2Du) = 0 .
Now, switching from the intrinsic cylinder Qz0(λ2−pR2, R) to Q0(1,
1), that is mak- ing the change of variables
v(x, t) := u(x0 +Rx, t0 + λ2−pR2t) (x, t) ∈ B1 × (−1, 1) ≡ Q1
,
we note that (3.31) gives that
(3.32) vt −4v = 0 ,
holds in the cylinder Q1. Therefore this argument tells us that on
an intrinsic cylinder of the type in (3.27) the solution u
approximately behaves as a solution to the standard heat system,
and therefore enjoys good estimates.
Note that for p = 2 the cylinders considered in (3.27) are actually
the standard parabolic ones - that is those equivalent to the balls
generated by the parabolic metric in Rn+1
dpar((x, t), (y, s)) := |x− y|+ |t− s| 12 , x, y ∈ Rn , s, t ∈ R
.
These are in turn independent of the solution, and therefore if
ones wants to derive CZ estimates for solutions to (3.24) in the
case p = 2, then the standard elliptic proof works, provided using
the parabolic maximal operator, that is the one defined by
considering as defining family the one parabolic cylinders
[Mparf ](x, t) := sup (x,t)∈Qr
∫ Qr
|f(y, s)| dy ds , Qr ≡ Br(x0)× (t0 − r2, t0 + r2) .
Instead, in the case p > 2 one is lead to consider intrinsic
cylinders as in (3.27) which depend on the solution themselves, and
therefore do not define a universal
16 GIUSEPPE MINGIONE
family. In a sense, we are considering the locally deformed
parabolic metric given by
(3.33) dpar,λ((x, t), (y, s)) := max |x− y|+ λ p−2 2 |t− s| 12
,
where, again the number λ depends on the solution via (3.28).
Remark 3.2 (Interpolation nature of Theorem 3.10). A closer look at
the proofs in [4, 56] reveals a more explicit structure of estimate
(3.30), which actually looks like (
− ∫ QR/2
) 1 p
] p 2
,
where the constant c1 depends on n,N, p, ν, L, but is independent
of q. Therefore, considering the case F ≡ 0 and eventually letting
γ → ∞ the previous estimate yields
(3.34) sup QR/2
|Du| ≤ c1 ( − ∫ QR
) 1 2
,
which is the original L∞-gradient estimate obtained by
DiBenedetto-Friedmann [47, 48] for solutions to the homogeneous
p-Laplacean system (3.12). This phenomenon reflects the
interpolation nature of Theorem 3.10, which in some sense provides
an estimate which interpolates the trivial Lp estimate - that is
(3.30) with γ = p, after absorbing the intermediate integral via
standard methods - which comes from testing the system with the
solution, and the L∞ one (3.34). A similar remark applies to the
case p < 2 treated a few lines below.
We turn now to the case p < 2. This is the so called singular
case since when |Du| approaches zero, the quantity |Du|p−2, which
roughly speaking represents the lowest eigenvalue of the operator
div (|Du|p−2Du), tends to infinity. Anyway, this interpretation is
somewhat misleadins here: we are interested in determining the
integrability rate of Du, therefore we are interested in the large
values of the gradient. Here a new phenomenon appears: we cannot
consider values of p which are arbitrarily close to 1, as described
in [46]. The right condition turns out to be
(3.35) p > 2n
n+ 2 ,
otherwise, as shown by counterexamples, solutions to (3.26) maybe
even unbounded. This can be explained by looking at (3.26) when
|Du| is very large: if p < 2, and it is far from 2, then the
regularizing effect of the elliptic part - the diffusion - is too
weak as |Du|p−2 is very small, and the evolutionary part develops
singularities like in odes, where no diffusion is involved.
For the case p < 2 the result is now
Theorem 3.11 ([4]). Let u ∈ C(0, T, L2(,RN ))∩Lp(0, T,W 1,p(,RN ))
be a weak solution to the parabolic system (3.24), where is a
bounded domain in Rn, and p < 2 satisfies (3.35). Then
F ∈ Lγloc(T ,RNn) =⇒ Du ∈ Lγloc(T ,RNn) for every γ ≥ p .
TOWARDS A NON-LINEAR CALDERON-ZYGMUND THEORY 17
Moreover, there exists a constant c ≡ c(n,N, p, ν, L, q) such that
for every parabolic cylinder QR ≡ BR(x0)× (t0 −R2, t0 +R2) b T it
holds that(
− ∫ QR/2
) 1 p
] 2p p(n+2)−2n
.
Note how in the previous estimate the scaling deficit exponent p/2
in (3.30) is replaced by 2p/(p(n+2)−2n), a quantity that stays
finite as long as (3.35) is satis- fied. Therefore estimate (3.36)
exhibits in quantitative way the role of assumption (3.35).
Theorems 3.10-3.11 admit of course several possible
generalizations; a first one concerns general parabolic equations
of the type
ut − div a(Du) = div (|F |p−2F ) ,
where the vector field a(·) satisfies (3.14). In this case Theorems
3.10-3.11 hold in the form described above, for a constant c
depending also on ν, L.
We shall now outline a couple of non-trivial extensions, recently
obtained in [56]. The first concerns the evolutionary p-Laplacean
system with coefficients
) .
The point here is that while the function depending on the space
variable c(x) is assumed to be VMO regular, that is to satisfy
assumptions (3.11), this time dependent measurable function b(t) is
assumed to satisfy only
0 < ν ≤ b(t) ≤ L <∞ ,
while no pointwise regularity is assumed other than the obvious
measurability. Under such assumptions for solutions to (3.37)
Theorem 3.10 holds exactly in the form presented above, but for the
fact that the constant c also depends on the function c(·). This
result, and the related one obtained in [56], is a far reaching
extension of recent analogous results due to Krylov [89] and his
students, who consider a similar situation in the case of linear
parabolic equations (when, in particular, p = 2, and no intrinsic
geometry needs to be considered).
The second result from [56] we are presenting is the parabolic
analog of Theorem 3.8, we are therefore treating general parabolic
systems of the type
ut − a(x, t,Du) = div (|F |p−2F ) ,
while the assumptions on a(·) are
(3.38)
1 2 |Da(x, t, z)| ≤ L(s2 + |z|2)
p−1 2
ν(s2 + |z|2) p−2 2 |λ|2 ≤ Da(x, t, z)λ, λ
|a(x, t, z)− a(x0, t, z)| ≤ Lω (|x− x0|) (s2 + |z|2) p−1 2 ,
whenever x, x0 ∈ , t ∈ (0, T ), z, λ ∈ RNn, where p ≥ 2 and s ∈ [0,
1]. Note that, again, we are assuming no continuity of t 7→ a(x, t,
z), this being just a measurable map. The result is finally
Theorem 3.12 ([56]). Let u ∈ C(0, T, L2(,RN )) ∩ Lp(0, T,W 1,p(,RN
)) be a weak solution to the general parabolic system (3.24), where
is a bounded domain in Rn, and p ≥ 2, N ≥ 1, and where the vector
field satisfies (3.38). Then there exists a positive number δ ≡
δ(n,N, p, ν, L) > 0 such that
F ∈ Lγloc(T ,RNn) =⇒ Du ∈ Lγloc(T ,RNn) ,
18 GIUSEPPE MINGIONE
n + δ .
Moreover, there exists a constant c ≡ c(n,N, p, ν, L, γ) such that
for every parabolic cylinder QR ≡ BR(x0)× (t0 −R2, t0 +R2) b T the
local estimate (3.30) holds.
3.7. Open problems. For the sake of brevity we shall restrict here
to the model equation (3.4). Let us start from one simple
observation. The minimum degree of integrability required to Du and
F in order to give meaning to the weak formulation of (3.4), that
is∫
is clearly given by
u ∈W 1,p−1(,RN ) and F ∈ Lp−1(,RNn) .
This leads to consider those distributional solutions to (3.4)
which do not belong to the natural space W 1,p, and therefore are
not energy solutions; these are very weak solutions. We shall
encounter such solutions also later on, when dealing with measure
data problems, and we shall see that they can exist beside the
usual energy solutions.
Now a comparison between the result of Theorem 3.2 and the linear
one in (2.12), which regards solutions to (2.11), naturally leads
to the following open problem, which is actually a conjecture of
Iwaniec:
Open problem 1. Prove that the results of Theorems 3.1 and 3.2 hold
in the full range p− 1 < γ.
The only result known up to now in this direction is due
independently to Iwaniec & Sbordone [72, 69], and Lewis [90],
who were able to prove that the statement of Theorem 3.2 holds in
the range p − ε < γ < ∞ for some ε > 0, depending on the
exponent p and the dimension n, but independent of all the other
entities considered, in particular of the solution. The methods of
proof in [72] uses Iwaniec’s non-linear Hodge decomposition, a
powerful and deep tool of its own interest. The method in [90]
relies instead on the truncation of maximal operators, the
so-called Lipschitz truncation method.
The conjecture above extends to solutions to the parabolic system
(3.24). Again, in this direction Kinnunen & Lewis [82, 83]
proved the validity of Theorems 3.10- 3.11 in the range p − ε <
γ < p + ε, thereby finding the right extension of so called
Gehring’s lemma [63, 70] in the case of parabolic problems.
Bogelein, in a remarkably deep paper [26], recently extended the
results of Kinnunen & Lewis on very weak solutions [83] to the
case of parabolic systems depending on higher order spatial
derivatives; see also [25], which features the higher order
extension of [82].
We close this section noting that the weaker integrability
assumption F ∈ Lγ with γ < p, puts the right hand side of (3.4)
outside the natural dual space
div (|F |p−2F ) 6∈W−1,p′() ,
and therefore leads us to consider those problems, as measure data
ones, for which we face the problem of proving gradient estimates
below the duality exponent.
3.8. Obstacle problems. We conclude with further integrability
results, recently obtained in [27], and concerning gradient
estimates for obstacle problems. A point of interest here is that,
differently from the usual results available in the literature, the
obstacles considered here are just Sobolev functions, and therefore
discontinu- ous, in general.
TOWARDS A NON-LINEAR CALDERON-ZYGMUND THEORY 19
We shall start with the simplest elliptic case, involving the
minimization problem
(3.40) Min
0 () : v ≥ ψ a.e.} ψ ∈W 1,p 0 () .
The integrability result available is
Theorem 3.13. Let u ∈ W 1,p() be the unique solution to the
obstacle problem (3.40), where is a bounded domain in Rn.
Then
ψ ∈W 1,γ loc () =⇒ u ∈W 1,γ
loc () for every γ ≥ p . Moreover, for every BR b it holds
that(
− ∫ BR/2
+ c
( − ∫ BR
,
where c ≡ c(n, p, ν, L, γ).
The previous theorem is obviously optimal, as it follows by
considering the gra- dient integrability of u on the contact set {u
≡ ψ}, where Du and Dψ coincide almost everywhere.
The next result concerns the evolutionary case; for the sake of
simplicity we shall confine ourselves to the case of time
independent obstacles and zero right hand side, referring to [27]
for the time dependent case and more general right hand sides. We
shall consider parabolic variational inequalities in the cylinder
(3.25), defined in the class
K = {v ∈ Lp(0, T ;W 1,p 0 ()) ∩ L2(T ) : v ≥ ψ a.e.}
where the exponent p is as usual assumed to satisfy (3.35), and ψ :
T → R is a fixed obstacle function that for simplicity we assume to
be time independent. The variational inequality under consideration
is
(3.41)
∫ T
vt(v − u) + |Du|p−2Du,D(v − u) dz + (1/2)v(·, 0)2L2() ≥ 0,
for all v ∈ K ∩W 1,p′(0, T ;W−1,p′()). Finally, for the initial
values of u we shall
assume that there exists v0 ∈ K ∩W 1,p′(0, T ;W−1,p′()) such
that
(3.42) u(x, 0) = v0(x, 0) for x ∈ .
The integrability result is now
Theorem 3.14. Let u ∈ K satisfy the integral inequality (3.41)
under the assump- tion (3.42) with p > 2n
n+2 . Then
Dψ ∈ Lγloc(T ) =⇒ Du ∈ Lγloc(T ) for every γ ≥ p . Moreover, there
exists a constant c = c(n, p, γ) such that for any parabolic
cylinder Q2R(z0) b T there holds(
− ∫ QR/2
) 1 p
]d ,
where
p(n+2)−2n if p < 2.
20 GIUSEPPE MINGIONE
Both in Theorem 3.13 and in Theorem 3.14 we have considered the
model case of the p-Laplacean operator, but more general operators
can be also considered; for this we again refer to [27]. In the
same paper one can also find conditions for treating time dependent
obstacles; in this case in order to get the Lγ local integrability
of the spatial gradient it is necessary to assume at least
that
∂tψ ∈ L γ p−1
loc (T ) .
4. Review on measure data problems
Following a rather consolidated tradition we shall talk about
measure data prob- lems also in those cases when the datum involved
is not genuinely a measure, but also a function with low
integrability properties. For the sake of simplicity we shall
concentrate on Dirichlet problems, with homogeneous boundary datum,
of the type
(4.1)
u = 0 on ∂,
where ⊂ Rn is a bounded open subset with n ≥ 2, while µ is a
(signed) Borel measure with finite total mass
|µ|() <∞ .
Non-homogeneous boundary data can be dealt with by standard
reductions, and will not be treated here. Of course, it is always
possible to assume that the measure µ is defined on the whole Rn by
letting µ(Rn \ ) = 0, therefore in the following we shall do so. As
for the structure properties of the problem, these are essentially
the standard ones prescribing growth and monotonicity properties at
p-rate: in the following a : × Rn → Rn will denote a Caratheodory
vector field satisfying
(4.2)
{ ν(s2 + |z1|2 + |z2|2)
p−2 2 |z2 − z1|2 ≤ a(x, z2)− a(x, z1), z2 − z1
|a(x, z)| ≤ L(s2 + |z|2) p−1 2 ,
for every choice of z1, z2 ∈ Rn, and x ∈ . Unless otherwise stated,
when consider- ing (4.2) the structure constants will satisfy
(4.3) 2 ≤ p ≤ n 0 < ν ≤ 1 ≤ L s ≥ 0 .
In particular we remark that at this stage x 7→ a(x, ·) is only a
measurable map. For this reasons many of the following results
readily extend to problems of the type
(4.4)
u = 0 on ∂,
but again for brevity we shall confine ourselves to simpler ones as
in (4.1). See for instance [106, Section 6.4] for the treatment of
such cases in the context of the results we are going to
present.
Again for brevity, we shall not deal with the sub-quadratic case 1
< p < 2, where additional problems appear. For this we refer
to [16, 51].
Now, let’s switch to the facts. How to prove the existence of a
solution? And, what kind of solutions we are talking about? This is
already an issue introduced in Section 3.7: here very weak
solutions come back. We start by the following crude distributional
definition, particularizing the one in (3.3)
Definition 4.1. A solution u to the problem (4.1) under assumptions
(4.2), is a
function u ∈W 1,1 0 () such that a(x,Du) ∈ L1(,Rn) and
(4.5)
∫
TOWARDS A NON-LINEAR CALDERON-ZYGMUND THEORY 21
Energy solutions form a class in which the unique solvability of
(4.1) is possible. In fact given u, v two energy solutions to
(4.1), we can test the weak formulation∫
a(x,Du)− a(x,Dv), D dx = 0
by = u− v, and then (4.2)1 and Poincare inequality implies that u ≡
v. As already mentioned in Section 3.7, solutions which are not
energy solutions are
usually called very weak solutions, and they exists beside usual
energy solutions, even for simple linear homogeneous equations of
the type
div (A(x)Du) = 0 ,
as shown by a classical counterexample of Serrin. In fact, in his
fundamental paper [111] Serrin showed that, for a proper choice of
the strongly elliptic and bounded, measurable matrix A(x), the
previous equation admits at least two solutions: one of them
belongs to the natural energy space W 1,2, and it is therefore an
energy solution; the other one does not belong to W 1,2, and for
this reason in a time where the concept of very weak solution was
not very familiar, was conceived as a pathological solution. This
situation immediately poses the problem of uniqueness of solutions.
In fact, the problem of finding a definition of solution allowing
for unique solvability of (4.1), similarly as what happens for
energy solutions, is in general still open, and is the most
important and outstanding one in the theory of measure data
problems. For a comprensive discussion on the uniqueness problem we
refer to [43].
Going back to the existence problem, in the case µ ∈ (W 1,p 0 ())′
≡ W−1,p′(),
the dual of W 1,p 0 (), then the standard monotone operator theory
[98] provides us
with the existence of a solution u ∈ W 1,p 0 , which is at this
point unique amongst
the energy solutions. The first issue is therefore to establish the
existence of a solution in the case µ 6∈
W−1,p′(). Let me examine a few related situations, also related to
the theorems in the following sections.
Example 1 (Measures with low density). In the case of a Borel
measure µ in the right hand side, there is a classical trace type
theorem due to Adams [5] stating that if the density
condition
(4.6) |µ|(BR) . Rn−p+ε
holds for some ε > 0, then it follows that µ ∈ (W 1,p 0 ())′ ≡
W−1,p′(). Therefore
for such measures we have the existence of a unique energy
solution. We notice that the p-capacity of a ball BR is comparable
to Rn−p, therefore (4.6) implies that the measure in question is
absolutely continuous with respect to the p-capacity. Indeed
Sobolev functions are those that can be defined up to set of
negligible p-capacity. For a general uniqueness theorem concerning
measures which are absolutely con- tinuous with respect to the
capacity without necessarily satisfying (4.6) we refer to [24];
this paper concerns so called entropy solutions, a kind of solution
we shall not discuss here.
Example 2 (High integrable functions). When the measure is a
function µ = f ,
which belongs to Lγ(), then for certain values of γ we have f ∈
W−1,p′(). In fact let me recall that Sobolev imbedding theorem
yields, when p < n
W 1,p 0 () ⊂ Lp
∗ () p∗ :=
n− p .
Therefore L(p∗)′() ⊂W−1,p′(). This means that if f ∈ Lγ() and
γ ≥ (p∗)′ = np
np− n+ p
22 GIUSEPPE MINGIONE
then there is a unique energy solution to (4.1). This argument can
be refined up to Lorentz spaces - see Definition 5.4 below. In fact
the improved Sobolev embedding theorem gives
W 1,p 0 () ⊂ L(p∗, p)() $ Lp
∗ () = L(p∗, p∗)() .
For a proof see for instance [119]. But since (L(p∗, p))′ =
L((p∗)′, p′), we have that if f ∈ L((p∗)′, q)() with q ≤ p′, then
there exists a unique energy solution to (4.1).
Example 3 (Non-linear Green’s functions). By the fundamental
solution to the p-Laplacean equation we mean the function
(4.7) Gp(x) ≈
log |x| if p = n .
Compare with (2.6).
Remark 4.1. With abuse of notation we shall go on denoting by Gp
any function differing by the one in (4.7) for a multiplicative or
an additive constant. This is why we used the symbol ≈ in
(4.7).
Denoting by δ the Dirac mass charging the origin, up to a
re-normalization constant depending only on n and p, the function
Gp(·) properly solves the measure data equation
(4.8) 4pu := div (|Du|p−2Du) = δ ,
see for instance [43]. It is interesting to compute the degree of
integrability of Gp(·). Note that
|DGp| ≈ |x| 1−n p−1 ,
and therefore by (2.21) it follows that
(4.9) |Gp|p−1 ∈M n n−p loc (Rn) and |DGp|p−1 ∈M
n n−1
loc (Rn) ,
the first being meaningful of course when p < n. The previous
inclusions should be compared with (2.20).
It is important to keep in mind (4.9), as well as the integrability
exponents thereby introduced, will serve as a reference for testing
the optimality of the regu- larity results presented later for
solutions to general non-linear problems as (4.1). Here we need a
clarification; in fact in Section 4.4 below we shall see that
Gp(·), or rather, a minor modification of it via an additive
constant to meet boundary values, is the unique solution to (4.8) -
with zero boundary value - in the class of solutions obtainable via
certain positive approximations - see next section.
For this reason Gp(·) is safely conjectured to be the unique
distributional so- lution to (4.8) amongst all possible entropy or
so called re-normalized solutions [43]. Moreover, as shown later in
Theorem 5.4 and Remark 5.3, Gp(·) exhibits the worst behavior
amongst the solutions with measure data problems, according to the
rough principle stating that “the more the measure concentrates,
the worse solutions behave”.
As a conclusion, in the following we shall use the regularity
properties of Gp(·) described in (4.9) to test the optimality of
the regularity results obtained for ap- proximate solutions to
problems involving general measure as (4.1).
4.1. Solvability. Here we discuss the basic solvability of (4.1).
The main point is that although uniqueness is still lacking, it is
always possible to solve (4.1) in the plain sense of Definition
4.1. Since distributional solutions are not unique, at this point
there are in the literature several definitions of solution adopted
towards the settlement of the uniqueness problem - see for instance
[16] for the definition of entropy solutions, and [43] for the
definition of re-normalized ones. Here we shall
TOWARDS A NON-LINEAR CALDERON-ZYGMUND THEORY 23
adopt the notion of solution obtained by limits of approximations
(SOLA); these are solutions obtained via an approximation scheme
using solutions of regularized problems. We adopt this notion for a
number of reasons: our main emphasis on the a priori regularity
estimates; all the notions of solutions at the end turn out to
provide uniqueness in the same cases i.e. when considering measure
which are L1-functions, and turn out to be essentially equivalent
in such cases; in any case the existence of any of such solutions
is constructed via an approximation procedure. There are anyway a
few advantages in considering such SOLA: first, when deriving
regularity estimates one is dispensed from tedious calculations
implied, in the cases of different definitions of solutions, by the
fact that one has to verify additional conditions when using test
functions; in the case of approximate solutions one essentially
argues on an a priori level, testing the equations as one would
have a usual energy solutions, as, in fact, the approximating
solutions are. Second, as already noted a few lines above, when
dealing with SOLA solutions it is easy to verify the uniqueness, in
its Dirichlet class, of the fundamental solution (4.7), and this
allows for claiming the optimality of the regularity results
obtained about SOLA; see Section 4.4 below. Moreover, as we shall
see later, in certain special cases, approximate solutions allows
to formulate additional uniqueness results; see Section 5.5
below.
The approximation procedure has been settled by Boccardo &
Gallouet [22, 23], see also [42]. The idea is to approximate the
measure µ via a sequence of smooth functions {fk} ⊂ L∞(), such that
fk → µ weakly in the sense of measures, or fk → µ strongly in L1()
in the case µ is a function. At this point, by standard
monotonicity methods, one finds a unique solution uk ∈W 1,p 0 ()
to
(4.10)
uk = 0 on ∂.
The arguments in [22] lead to establish that there exists u ∈W
1,p−1 0 () such that,
up to a not relabeled subsequence,
(4.11) uk → u and Duk → Du strongly in Lp−1(), and a.e.
and (4.1) is solved by u in the sense of (4.5). We have therefore
found a distribu- tional solution having the remarkable additional
feature of having been selected via an approximation argument
through regular energy solutions.
It is important to notice that, as described for instance in [20,
42], in the case µ is an L1-function, by considering a different
approximating sequence {fk} strongly converging to f in L1(), we
still get the same limiting solution u. As a consequence the
described approximation process allows to build a class of
solutions, those in fact obtained by approximation, in which the
unique solvability of (4.1) is possible. Related uniqueness
properties follows for entropy solutions when the measure is a
function; see [24].
For the reason we have just explained, from now on, when dealing
with the case the measure µ is a actually an L1-function, we shall
talk about the solution to (4.1), meaning by this the unique
solution found by the above settled approximation scheme.
4.2. Basic regularity results. There is a vast literature on the
regularity of so- lutions to measure data problems. We shall
confine ourselves to the first papers dealing with general
quasilinear problems of the type (4.1). The basic regularity
results obtained fall in two categories; the first deals with
genuine measure data problems and in some sense reproduce for
general solutions the integrability proper- ties of the fundamental
solution (4.9). The second deals with the case the measure
24 GIUSEPPE MINGIONE
is a function enjoining extra integrability properties, and aims at
reproducing in the non-linear case the CZ linear results of the
type (2.4).
Theorem 4.1 ([16, 52]). Under the assumptions (4.2) there exists a
solution u ∈ W 1,p−1
0 () such that
|u|p−1 ∈M n n−p () for n < p ,
and
(4.12) |Du|p−1 ∈M n n−1 () .
The result of the previous theorem has been obtained in some
preliminary forms in [22, 117]; the form above has been obtained in
[16] for the case p < n, while the case p = n, with the
consequent Mn estimate, is treated in [52]. Results for systems
have been obtained in [50].
We now switch to the case when the measure is actually a
function
(4.13) µ ∈ Lγ() , γ ≥ 1 .
For this we premise the following:
Remark 4.2 (Maximal regularity). The equations we are considering
have mea- surable coefficients, and this means that x 7→ a(x, z) is
a measurable map. The maximal regularity in terms of gradient
integrability we may expect, even for en- ergy solutions to the
homogeneous equation div a(x,Du) = 0, is at most Du ∈ Lqloc, for
some q which is in general only slightly larger than p, and depends
in a criti- cal way on n, p, L/ν. This is basically a consequence
of Gehring’s lemma [63, 70]. Therefore we are not expecting to get
much more that Du ∈ Lp in general for so- lutions to the measure
data problems considered in the following. Therefore, with abuse of
terminology, we shall consider Du ∈ Lp as the maximal regularity
for the gradient of solutions u.
The previous remark allows to restrict the range of parameters of
γ, the exponent appearing in (4.13). We first look for values of γ
such that Lγ 6⊂W−1,p′ , otherwise the existence of an energy
solution such Du ∈ Lp follows. We are in fact almost at the maximal
regularity. As a matter of fact when considering measure data
problems one is mainly interested in those solutions which are not
energy ones. By Example 2 we see that the right condition is
(4.14) 1 < γ < np
np− n+ p = (p∗)′ for p < n .
Theorem 4.2 ([23]). Under the assumptions (4.2) and (4.13)-(4.14),
the solution
u ∈W 1,p−1 0 () to (4.1) is such that
|Du|p−1 ∈ L nγ n−γ () .
Finally, a borderline case
Theorem 4.3 ([23]). Assume that (4.2) hold and that the measure µ
is a function
belonging to L logL(). The solution u ∈W 1,p−1 0 () to (4.1) is
such that
|Du|p−1 ∈ L n n−1 () .
Theorems 4.1-4.3 establish a low order CZ-theory for elliptic
problems with mea- sure data which is completely analogous to that
available in the linear case and therefore optimal in the scale of
Lebesgue’s spaces - compare with (2.3) and the other results in
Section 2.
We just talked about a low-order theory since, when dealing with
second order equations, one would expect to get results on second
order derivatives or the like, as
TOWARDS A NON-LINEAR CALDERON-ZYGMUND THEORY 25
in (2.3) for the linear case. Indeed, in the next sections we shall
extend Theorems 4.1-4.3 in two different directions. We shall in
fact present
1) Integrability theorems for the gradient in function spaces
different from the Lebesgue’s ones. In particular, we shall
consider non rearrangement invariant func- tion spaces (Section
5).
2) Differentiability theorems for the gradient, having in turn,
when dealing with more regular equations, weak forms of Theorems
4.1-4.3 as a corollary (Section 6).
4.3. Open problems. There are in our opinion at least two main open
issues in the theory of measure data problems. The first has been
already mentioned:
Open problem 2. Find a functional class where problem (4.1) can be
uniquely solved; the solution found must be distributional.
The second open problem is concerned with a tremendous gap in the
theory of measure data problems, as it is far mote important from
the regularity theory viewpoint.
Open problem 3. Prove solvability of the Dirichlet problem (4.1)
under assump- tions (4.2) in the case (4.1)1 is a system, and u
takes its values in RN , N > 1; in particular prove Theorems
4.1-4.3 in the case of systems.
A weaker version of the previous one is
Open problem 4. Find classes of structure conditions on the vector
field a : × RNn → RNn, still satisfying (4.2), allowing for
solvability of the Dirichlet problem (4.1) in the case of
systems.
The first attempts to prove the last two problems are in [50, 51,
52, 60]. Such papers give a complete solution for the case of
certain systems with a special struc- ture, first introduced by
Landes [91], and aimed at selecting those vector fields a(·) whose
ellipticity properties match with the gradient of certain truncated
maps in the vectorial case. Such condition is satisfied for
instance by systems depending on the gradient in a peculiar way;
the so called Uhlenbeck structure, first considered in [121] to
select homogeneous systems allowing for everywhere regular
solutions
(4.15) a(Du) ≡ b(|Du|)Du , b(t) ≈ tp−2
is a typical example. The important case of the p-Laplacean system
with measure data
4pu := div (|Du|p−2Du) = µ
is therefore covered. The difficulties involved in the Open problem
3, are of two types, and they are in turn related each other. The
main point is that due to the presence of a measure in the weak
formulation (4.5) we can only allow L∞- functions, and therefore in
order to prove energy estimates one is led to test (4.5) with
truncations of the solutions:
w(x) := max{−k,min{u(x), k}} k ∈ N .
It turns out that while in the scalar case the gradient of w is
justDuχ{−k≤u(x)≤k}, in the vectorial case it has a more delicate
expression which does not fit the ellipticity properties of a
general system, but in some special cases, as, in fact, the one in
(4.15). This obstruction pops-up both at the level of getting a
priori regularity estimates, and at the level of the convergence
estimates, that is those related to (4.11); for more details see
[50].
26 GIUSEPPE MINGIONE
4.4. On the uniqueness of fundamental solutions. Here we shall
present a uniqueness result for the Dirichlet problem
(4.16)
{ −4pu = δ in B1
u = 0 on ∂B1 ,
which is known amongst experts. More precisely we shall see that -
compare with the notation introduced in (4.7) and subsequent remark
- the non-linear fundamen- tal solution
(4.17) Gp(x) ≈
) if 2 ≤ p < n
log |x| if p = n ,
is the unique solution to the problem (4.16) amongst those
obtainable via approxi- mation, as described in (4.10), with
non-negative functions fk - compare with the calculations made in
[43, Example 2.16] where it is shown that for a suitable choice of
the constant involved in the symbol ≈ in (4.17), Gp is also a
re-normalized solution to (4.16).
This uniqueness result immediately identifies the function class
where to solve, in a unique way, problem (4.16), since the standard
way to to build a sequence of functions fk weakly converging to µ
in the sense of measures, is obviously via convolutions with
standard mollifiers {φk}
(4.18) fk := µ ∗ φk ,
which provide positive approximations data fk when µ is a
non-negative measure. As a matter of fact, when considering the
approximation scheme (4.10) one always uses the approximations
settled in (4.18), that therefore should be considered as a part of
the approximation scheme in (4.10), and therefore of the
definitions of approximate solutions (SOLA).
We explicitly point out that the following argumentation
singles-out in a very clear way the difference between any
distributional solution, and a solution obtained as a limit of
approximations.
We start by considering the approximating problems to (4.16),
which, according to (4.10), are now defined by{
−4puk = fk in B1
u = 0 on ∂B1 .
(4.19) uk → u and Duk → Du
strongly in Lp−1 and almost everywhere. This result follows from
[22, 23], as already explained in Section 4.1. As a
consequence:
(4.20) u ∈W 1,p−1(B1) .
We want to show that
(4.21) u ≡ Gp and now Gp is defined in (4.17).
By the very definition (4.18) it follows that fk ≥ 0, and since fk
is a smooth function we can apply the maximum principle to the
energy solution uk, thereby getting that
(4.22) uk ≥ 0
since uk ≡ 0 on ∂B1. Combining (4.19) and (4.22) we infer
(4.23) u ≥ 0 ,
TOWARDS A NON-LINEAR CALDERON-ZYGMUND THEORY 27
almost everywhere in B1. We now get further regularity properties
of u. We have that, in a distributional sense
4pu = 0 in B1 \ {0}
and therefore standard regularity theory for solutions to the
p-Laplacean equation tells that u is continuous in B1 \ {0}; we
have actually more, and this will be important in a few
lines:
(4.24) u ∈ C1,α loc (B1 \ {0}) ,
for some α ∈ (0, 1). In particular, we have that (4.23) holds
everywhere. As a corollary we also have
(4.25) Du ∈ Lp(B1 \Br) for every r ∈ (0, 1) .
Using this fact together with (4.23) we may apply a classical
result of Serrin [112, Theorem 12] to infer that there exists a
positive constant c such that
(4.26) 1
c ≤ u
Gp ≤ c .
Now we recall the following result:
Theorem 4.4 ([75]). There exists a unique distributional solution
to the problem (4.16) u such that: u ∈ C1(B1 \ {0}), Du ∈ Lp−1(B1),
Du ∈ Lp(B1 \Br) for every r ∈ (0, 1), and satisfying condition
(4.26).
Since u ∈W 1,p−1(B1), by (4.20), (4.24), (4.25) and (4.26) all the
assumptions of the previous theorems are satisfied by u, and being
obviously satisfied also by Gp, (4.21) finally follows.
5. Nonlinear Adams theorems
The results in this section follow simultaneously two different
viewpoints: first, they extend in new directions the regularity
results available for measure data prob- lems presented in Section
4, showing CZ estimates in new types of function spaces; second,
and more importantly, they show that certain classical potential
theory facts apparently linked to the linear setting, can be
actually reformulated in the context of what is called non-linear
potential theory. We shall in fact present optimal non- linear
extensions of classical results of Adams [6] and Adams & Lewis
[9]. Moreover, we shall present a localization of the classical
Lorentz spaces estimates obtained by Talenti [117]. All the results
presented in this and in the next section are taken from [105,
106].
Remark 5.1. When stating our regularity results for measure data
problems, we shall state them in the form of existence and
regularity results, since the uniqueness of distributional
solutions does not hold. On the other hand the regularity claimed
holds for every approximate solution to (4.1) in the sense of the
approximation scheme described in Section 4.1.
5.1. Morrey spaces. Morrey spaces provide a way of measuring the
size of func- tions which is in some sense orthogonal to that of
Lebesgue spaces. In fact, while these read the size of the
super-level sets of functions - as rearrangement invariant spaces -
Morrey spaces use instead density conditions in their formulation.
Specif- ically, the condition is
(5.1)
∫ BR
|w|γ dx ≤MγRn−θ and 0 ≤ θ ≤ n ,
to be satisfied for all balls BR ⊆ with radius R.
28 GIUSEPPE MINGIONE
Definition 5.1. A measurable map w : → Rk, belongs to the Morrey
space Lγ,θ(,Rk) ≡ Lγ,θ() iff satisfies (5.1), and moreover one
sets
wγ Lγ,θ()
Rθ−n ∫ BR
|w|γ dx .
Obviously Lγ,n ≡ Lγ , and Lγ,0 ≡ L∞. The Morrey scale is orthogonal
to the one provided by Lebesgue spaces in fact
(5.2) Lγ,θ ⊂ Lγ \ Lγ logL for every θ > n ,
that is, no matter how close θ is to zero, therefore no matter how
close Lγ,θ is to L∞ in the Morrey scale. We recall that the space
Lγ logL is that of those functions w satisfying ∫
|w|γ log (e + |w|) dx <∞ ,
so that Lγ ′ ⊂ Lγ logL ⊂ Lγ for every γ′ > γ ≥ 1.
In a similar way the classical Marcinkiewicz-Morrey spaces [9, 7,
109, 114] are naturally defined.
Definition 5.2. A measurable map w : → Rk belongs to the
Marcinkiewicz- Morrey space Mγ,θ(,Rk) ≡Mγ,θ() iff
sup BR⊂
sup λ>0
=: wγMγ,θ() <∞ .(5.3)
Extending in a endpoint way previous results of Stampacchia [114],
Adams proved the following extension of Theorem 2.1:
Theorem 5.1 ([6]). Let β ∈ [0, θ); for every γ > 1 such that βγ
< θ we have
(5.4) f ∈ Lγ,θ(Rn) =⇒ Iβ(f) ∈ L θγ
θ−βγ ,θ(Rn) .
(5.5) f ∈ L1,θ(Rn) =⇒ Iβ(f) ∈M θ
θ−β ,θ(Rn) ,
θ−β ,θ(Rn) .
In other words, we formally obtain Theorem 5.1 by Theorem 2.1 by
replacing n with θ. In fact, as noted above, Lγ,n ≡ Lγ . As shown
in [6] the embedding spaces of Theorem 5.1 are optimal with respect
to every scale. In particular Iβ(f) in general does not belong to
Lq() for any q > θγ/(θ − βγ).
Theorem 5.1 has a few immediate consequences. Via the
representation formula |u(x)| . I1(|Du|)(x) we have the so called
Sobolev-Morrey embedding theorem:
(5.7) Du ∈ Lγ,θ =⇒ u ∈ L θγ θ−γ ,θ provided 1 < γ < θ .
Along the same line there are applications to the regularity of
solutions to the Poisson equation (2.1):
(5.8) f ∈ Lγ,θ =⇒ Du ∈ L θγ θ−γ ,θ provided 1 < γ < θ .
TOWARDS A NON-LINEAR CALDERON-ZYGMUND THEORY 29
5.2. Nonlinear versions. Here we shall introduce the non-linear
potential theory versions of Theorem 5.1 and of (5.8) in the
context of measure data problems: this means that images of Riesz
potentials are replaced by solutions to non-linear equations with
p-growth, for example p-harmonic functions. Specifically, we are
dealing with solutions to problems of the type (4.1); due to the
fact that x → a(x, z) is just measurable, the maximal regularity
expected in terms of gradient integrability is essentially Du ∈ Lp;
see also Remark 4.2. By the discussion on the existence/uniqueness
to solutions to (4.1) made in Section 4.1, in the following, when
the measure µ will be a function we shall talk about “the solution”
meaning the one defined by approximation according to the scheme
described in Section 4.1, since in this case approximate solutions
are unique. In the case µ is a genuine measure regularity results
stated will refer to the found solution - compare with Remark
5.1.
Since after Theorem 5.1 we expect the Morrey space parameter θ to
play the role of the dimension n, in formal accordance with (4.14),
we start assuming that
(5.9) 1 < γ ≤ θp
θp− θ + p and p < θ ≤ n ,
a condition whose actual role will be discussed in Remark 5.2
below. We have the following non-linear potential theory version of
(5.4):
Theorem 5.2 ([106]). Assume (4.2), and that the measure µ is a
function belonging
to Lγ,θ() with (5.9). Then the solution u ∈ W 1,p−1 0 () to the
problem (4.1) is
such that
loc () .
Moreover, the local estimate
|Du|p−1 L
θγ θ−γ ,θ(BR/2)
≤ cR θ−γ γ −n(|Du|+ s)p−1L1(BR) + cfLγ,θ(BR)
holds for every ball BR ⊆ , with a constant c only depending on n,
p, L/ν, γ.
Observe that, on one hand for p = 2 inclusion (5.10) locally gives
back (5.8), while on the other (5.10) is also the natural Morrey
space extension of the non- linear result of Theorem 4.2, to which
it locally reduces for n = θ. In the relevant
borderline case γ = θp/(θp−θ+p) the maximal regularity Du ∈
Lp,θloc() ⊂ Lploc() holds.
Remark 5.2 (Sharpness of condition (5.9)). The parameters choice in
(5.9) is optimal for the gradient integrability in the sense that
the upper bound for γ is the minimal one allowing for the maximal
regularity Du ∈ Lp. In fact we have
(5.11) γ ≤ θp
θ − γ ≤ p .
Related to this fact is Theorem 5.3 below. Moreover, by Example 1
in Section 4 a Borel measure µ satisfying (4.6) for some ε > 0
and for every ball BR ⊂ Rn, belongs
to the dual space W−1,p′ , and therefore (4.1) is uniquely solvable
in W 1,p 0 (). This
is the reason for assuming p < θ in Theorem 5.2 and Theorem 5.4
below. The case p = θ forces γ = 1 and therefore falls in the realm
of measure data problems: it will be treated in Theorem 5.5 below.
Note that Holder’s inequality and (5.1) imply that |µ|(BR) = [|f |
dx](BR) . MRn−θ/γ , and therefore, again by the mentioned Adams’
result, in order to avoid trivialities we should also impose that
pγ ≤ θ. But keep in mind (5.9) and note that
(5.12) θp
30 GIUSEPPE MINGIONE
Therefore, assuming the first inequality in (5.11) together with p
≤ θ implies pγ ≤ θ.
Theorem 5.3 ([106]). Assume (4.2), and that the measure µ is a
function belonging
to Lγ,θ() with γ > θp/(θp−θ+p) and p ≤ θ ≤ n. Then the solution
u ∈W 1,p−1 0 ()
to the problem (4.1) is such that
Du ∈ Lh,θloc (), for some h ≡ h(n, p, L/ν, γ, θ) > p .
Moreover, for every ball BR ⊆ with R ≤ 1 the local estimate
DuLh,θ(BR/2) ≤ cR θ h−
n p−1 (|Du|+ s)Lp−1(BR) + cf
1 p−1
Lγ,θ(BR)
holds for a constant c only depending on n, p, L/ν.
In the case γ = 1 we cannot obviously expect Theorem 5.2 to hold;
instead, imposing an L logL type integrability condition on f
allows to deal with the case γ = 1 too, obtaining the natural
analog of (5.6).
Theorem 5.4 ([106]). Assume that (4.2) holds, and that the measure
µ is a function belonging to L1,θ() ∩