On Closed Boundary Value Problems for Equations of Mixed ...payne/cpam07-reprint.pdf · boundary value problems for linear partial differential equations of second order of the form

On Closed Boundary Value Problems for Equations

of Mixed Elliptic-Hyperbolic Type

DANIELA LUPOPolitecnico di Milano

CATHLEEN S. MORAWETZCourant Institute

AND

KEVIN R. PAYNEUniversità di Milano

To Louis Nirenberg on his 80th birthday

Abstract

For partial differential equations of mixed elliptic-hyperbolic type we prove re-

sults on existence and existence with uniqueness of weak solutions for closed

boundary value problems of Dirichlet and mixed Dirichlet-conormal types. Such

problems are of interest for applications to transonic flow and are overdetermined

for solutions with classical regularity. The method employed consists in variants

of the a − b − c integral method of Friedrichs in Sobolev spaces with suitable

weights. Particular attention is paid to the problem of attaining results with a

minimum of restrictions on the boundary geometry and the form of the type

change function. In addition, interior regularity results are also given in the im-

portant special case of the Tricomi equation. c© 2006 Wiley Periodicals, Inc.

1 Introduction

The purpose of this work is to examine the question of well-posedness of

boundary value problems for linear partial differential equations of second order

of the form

Lu = K (y)uxx + uyy = f in �,(1.1)

Bu = g on ∂�,(1.2)

where K ∈ C1(R2) satisfies

(1.3) K (0) = 0 and yK (y) > 0 for y �= 0.

� is a bounded, open, and connected subset of R2 with piecewise C1 boundary,

f and g are given functions, and B is some given boundary operator. We assume

throughout that

(1.4) �± := � ∩ R2± �= ∅,

Communications on Pure and Applied Mathematics, Vol. LX, 1319–1348 (2007)c© 2006 Wiley Periodicals, Inc.

1320 D. LUPO, C. S. MORAWETZ, AND K. R. PAYNE

so that (1.1) is of mixed elliptic-hyperbolic type. We will call � a mixed domain

if (1.4) holds. Additional hypotheses on f , g, K , and � will be given as needed.

Such an equation is of Tricomi type and goes by the name of the Chaplygin or

Frankl′ equation due to its longstanding importance in the problem of transonic

fluid flow. More precisely, equations of the form (1.1) describe the flow in the

hodograph plane where u represents either a stream function or a perturbation of

the velocity potential, and the variables x and y represent the flow angle and a

rescaled flow speed, respectively (see the classic monograph [4] or the modern

survey [22], for example). This connection is our principal motivation.

Such a boundary value problem will be called closed in the sense that the

boundary condition (1.2) is imposed on the entire boundary as opposed to an open

problem, in which (1.2) is imposed on a proper subset � ⊂ ∂�. Both kinds of prob-

lems are interesting for transonic flow; for example, open problems arise in flows

in nozzles and in proving nonexistence of smooth flows past airfoils, and closed

problems arise in constructing smooth transonic flows about airfoils; see [19]. It

is reasonable to conjecture that the inevitable shock that arises in the perturbation

of a smooth transonic flow springs from the singularity at the parabolic boundary

point in the closed problem; see [22]. Much more is known about open problems,

beginning with the work of Tricomi [30], who considered the case K (y) = y with

the boundary condition Bu = u on � = σ ∪ AC where σ is a simple arc in the

elliptic region y > 0 and AC is one of two characteristic arcs that form the bound-

ary in the hyperbolic region y < 0. Both the equation and this boundary condition

now carry his name.

On the other hand, much less is known about closed problems, partly because

closed problems for mixed-type equations are typically overdetermined for classi-

cal solutions. This is essentially due to the presence of hyperbolicity in the mixed-

type problem; for example, it is well-known that the Dirichlet problem for the

wave equation is generally not well-posed for classical solutions as first noted by

Picone [24] (see also the work of John [13] and Fichera [7]). More precisely, under

mild assumptions on the function K and the geometry of the boundary, one has a

uniqueness theorem for regular solutions to the Tricomi problem of the following

form: Let u ∈ C2(�)∩C1(�\{A, B})∩C0(�) solve Lu = K (y)uxx +uyy = 0 in

� and u = 0 on σ ∪ AC ; then u = 0 in all of �. Such uniqueness theorems have

been proven by a variety of methods, including energy integrals as in [26] and max-

imum principles as in [1, 18]. Such uniqueness theorems imply that the trace of a

regular solution u on the boundary arc BC is already determined by the boundary

values on the remainder of the boundary and the value of L(u) in �. Hence, if one

wants to impose the boundary condition on all of the boundary, one must expect

in general that some real singularity must be present. Moreover, in order to prove

well-posedness, one must make a good guess about where to look for the solution;

that is, one must choose some reasonable function space that admits a singularity

strong enough to allow for existence but not so strong as to lose uniqueness. This,

in practice, has proven to be the main difficulty of the problem.

CLOSED BVPS FOR MIXED-TYPE EQUATIONS 1321

Despite the interest in closed problems for mixed-type equations, the literature

essentially contains only two results on well-posedness. The first, due to Morawetz

[21], concerns the Dirichlet problem for the Tricomi equation (K (y) = y), and the

second due to Pilant [25] concerns the natural analogue of the Neumann problem

(conormal boundary conditions) for the Lavrentiev-Bitsadze equation (K (y) =

sgn(y)). In both cases, the restrictions on the boundary geometry are quite severe

as the domains must be lenslike and thin in some sense. Such restrictions on the

boundary geometry and the type change function are not particularly welcome in

the transonic flow applications since the boundary geometry reflects profile or noz-

zle shape and the approximation K (y) ∼ y is valid only for nearly sonic speeds.

The main purpose of this paper is to show well-posedness continues to hold for

classes of type change functions and for more general domains.

A brief summary of the results obtained follows. In Section 2, we investigate

the notions of distributional and strong solutions for the Dirichlet problem. In a

very general setting, we show both the existence of a distributional solution and

the uniqueness of the strong solution. However, the distributional solution may

neither be unique nor satisfy the boundary condition in a strong sense while the

strong solution may not exist. Hence, there is a need for a suitable intermediary

notion of solution to establish well-posedness. In Section 3, we define this suitable

intermediary notion and show that under slightly stronger restrictions on the type

change function and the domain, one has the existence of a unique weak solution

for the Dirichlet problem. Under similar restrictions, in Section 4, we show the

existence and uniqueness of a weak solution to a problem with mixed boundary

conditions (Dirichlet conditions on the elliptic boundary and conormal conditions

on the hyperbolic boundary).

The main ingredients in the proofs are global energy estimates obtained by vari-

ants of the classical a − b − c method of Friedrichs, also known as the multiplier

method, in which our multipliers are calibrated to an invariance or almost invari-

ance of the differential operator. In particular, Section 2 employs the technique

as first used by Protter [26] with a differential multiplier, Section 3 uses the idea

of Didenko [5] with an integral multiplier, and Section 4 uses the technique of

Morawetz [20] by first reducing the problem to a first-order system and then us-

ing a multiplier matrix. We note that the system technique has the advantages of

eliminating an order of differentiation and solving a more general problem (when

it works), but it has the disadvantage that one loses a degree of freedom in the

choice of the multiplier coefficients. The added degree of freedom when treating

directly the second-order scalar equation will be crucial in our treatment. Attempts

to improve Pilant’s work on the full conormal problem are in progress.

Finally, an analysis of regularity of the weak solutions is begun in Section 5,

where it is shown that the solutions have locally square-integrable first derivatives

in the interior and that the same is true for the second derivatives if one assumes that

the forcing term is slightly more regular. For simplicity, we treat explicitly only the


important special case of the Tricomi equation, avoiding the need to invoke any mi-

crolocal analysis. Instead, we exploit elementary estimates for sufficiently regular

approximate solutions obtained by mollifying with respect to the single variable x .

This single-variable variant of the well-known technique due to Friedrichs [8] for

showing that interior solutions are strong has been exploited by Lax and Phillips

[15] for extending Friedrichs’ result to the boundary and by Morawetz [21] in the

hyperbolic region for the Tricomi equation for obtaining regularity properties.

We conclude with a few additional remarks. While the existence and unique-

ness results of Section 3 for the Dirichlet problem do require some smoothness

assumptions on the boundary, no geometric assumptions on the geometry of the

elliptic boundary an arbitrarily small distance away from the parabolic line need to

be made. This results from an overlapping method for obtaining the needed a priori

estimates as described in Theorem 3.4. Similar considerations have been exploited

for the Tricomi problem (cf. [5, 16]) and will hold for other problems with an open

boundary condition, such as the Frankl′ problem. We also note that the arguments

used in obtaining the interior regularity results also apply for problems with open

boundary conditions as well. Finally, a remark about terminology is in order. For

the Soviet school of Berezanskiı̆ and Didenko, in the context of problems with open

boundary conditions, what we have called distributional and weak solutions were

called weak and generalized solutions, respectively.

2 Distributional and Strong Solutions for the Dirichlet Problem

In this section we show that even though closed boundary value problems for

mixed-type equations are overdetermined for classical solutions, at least for the

Dirichlet boundary condition, one can obtain a general existence result for a very

weak formulation of the problem as well as a general uniqueness result for a suf-

ficiently strong formulation. In particular, very little is assumed about boundary

geometry, and only mild and reasonable hypotheses are placed on the type change

function K .

In what follows, � will be a bounded mixed domain (open, connected, and

satisfying (1.4)) in R2 with piecewise C1 boundary so that we may apply the di-

vergence theorem. The function K ∈ C1(R) will be taken to satisfy (1.3), and

additional assumptions as necessary. We will make use of several natural spaces of

functions and distributions, namely, for treating the Dirichlet problem

Lu = K (y)uxx + uyy = f in �,(2.1)

u = 0 on ∂�.(2.2)

We define H 10 (�; K ) as the closure of C∞

0 (�) (smooth functions with compact

support) with respect to the weighted Sobolev norm

(2.3) ‖u‖H1(�;K ) :=

[ ∫�

(|K |u2x + u2

y + u2)dx dy

]1/2

.


Since u ∈ H 10 (�; K ) vanishes weakly on the entire boundary, one has a Poincarè

inequality: there exists CP = CP(�, K )

(2.4) ‖u‖2L2(�)

≤ CP

∫�

(|K |u2x + u2

y)dx dy, u ∈ H 10 (�; K ).

The inequality (2.4) is proven in the standard way by integrating along segments

parallel to the coordinate axes for u ∈ C10(�) and then using continuity. An equiv-

alent norm on H 10 (�; K ) is thus given by

(2.5) ‖u‖H10 (�;K ) :=

[∫�

(|K |u2x + u2

y)dx dy

]1/2

.

We denote by H−1(�; K ) the dual space to H 10 (�; K ) equipped with its negative

norm in the sense of Lax [14]

(2.6) ‖w‖H−1(�;K ) := sup0 �=ϕ∈C∞

0 (�)

|〈w, ϕ〉|

‖ϕ‖H10 (�;K )

,

where 〈 · , · 〉 is the duality bracket and one has the generalized Schwarz inequality

(2.7) |〈w, ϕ〉| ≤ ‖w‖H−1(�;K )‖ϕ‖H10 (�;K ), w ∈ H−1(�; K ), ϕ ∈ H 1

0 (�; K ).

One clearly has a rigged triple of Hilbert spaces

H 10 (�; K ) ⊂ L2(�) ⊂ H−1(�; K ),

where the scalar product (on L2, for example) will be denoted by ( · , · )L2(�). Other

notation and function spaces will be introduced as needed.

It is routine to check that the second-order operator L in (2.1) is formally self-

adjoint when acting on distributions D′(�) and gives rise to a unique continuous

and self-adjoint extension

(2.8) L : H 10 (�; K ) → H−1(�; K ).

Using standard functional analytic techniques, one can obtain results on distri-

butional existence and strong uniqueness for solutions to the Dirichlet problem

(2.1)–(2.2). The key point is to obtain a suitable a priori estimate by performing

an energy integral argument with a well-chosen multiplier.

LEMMA 2.1 Let � be any bounded region in R2 with piecewise C1 boundary. Let

K ∈ C1(R) be a type change function satisfying (1.3) and

K ′ > 0,(2.9)

∃δ > 0 such that 1 +

(2K

K ′

)′

≥ δ.(2.10)

Then there exists a constant C1(�, K ) such that

(2.11) ‖u‖H10 (�;K ) ≤ C1‖Lu‖L2(�), u ∈ C2

0(�).


PROOF: To obtain the estimate, one considers an arbitrary u ∈ C20(�) and a

triple (a, b, c) of sufficiently regular functions to be determined. One seeks to

estimate the expression (Mu, Lu)L2 from above and below where Mu = au +

bux + cuy is the as-yet-undetermined multiplier. One has that

(2.12) (Mu, Lu)L2 =

∫�

[div(Mu(K ux , uy)) − ∇(Mu) · (K ux , uy)]dx dy,

which can be rewritten with the aid of the divergence theorem as

(Mu, Lu)L2

=1

2

∫�

[αu2x + 2βux uy + γ u2

y + u2La] dx dy

+1

2

∫∂�

[2Mu(K ux , uy) − (K u2x + u2

y)(b, c) − u2(K ax , ay)] · ν ds,

(2.13)

where

α := −2K a − K bx + (K c)y,

β := −K cx − by,

γ := −2a + bx − cy,

ν is the unit exterior normal, and ds is the arc length element. For u with compact

support, the boundary integrals will vanish and the choices

(2.14) a ≡ −1, b ≡ 0, c = c(y) = max

{0,−

4K

K ′

}yield

(Mu, Lu)L2

=

∫�+

(K u2x + u2

y)dx dy +

∫�−

[1 +

(2K

K ′

)′](−K u2

x + u2y)dx dy

≥ min{1, δ}

∫�

(|K |u2x + u2

y)dx dy, u ∈ C20(�),

(2.15)

where �± := � ∩ {(x, y) : ±y ≥ 0}. Technically, since c is only piecewise

C1(�), one should first cut the integral along y = 0 and note that the boundary

terms (as given in (2.13)) along the cut will cancel out. We also point out that

the multiplier (a, b, c) defined in (2.14) is the average of the multiplier used by

Protter [26] for the Tricomi problem and the natural multiplier corresponding to

the adjoint problem.


Using the Cauchy-Schwarz inequality and the regularity and monotonicity of K ,

one finds the upper bound

(Mu, Lu)L2 ≤ ‖Mu‖L2(�)‖Lu‖L2(�)

≤ CK ‖u‖H10 (�;K )‖Lu‖L2(�), u ∈ C2

0(�),(2.16)

where CK depends on sup(|K/K ′|). To finish the estimate, one combines (2.15)

and (2.16) with the Poincarè inequality (2.4) to find (2.11) with C1(�, K ) =

[CK (1 + CP)/(min{1, δ})]1/2. �

An existence theorem for distributional solutions follows from the a priori esti-

mate (2.11), which is also an estimate for the formal transpose LT since LT = L .

THEOREM 2.2 Let � be any bounded region in R2 with piecewise C1 boundary.

Let K ∈ C1(R) be a type change function satisfying (1.3), (2.9), and (2.10). Then

for each f ∈ H−1(�; K ) there exists u ∈ L2(�) that weakly solves (2.1)–(2.2) in

the sense that

(2.17) (u, Lϕ)L2 = 〈 f, ϕ〉, ϕ ∈ H 10 (�; K ) : Lϕ ∈ L2(�).

PROOF: The basic idea is to combine the a priori estimate with the Hahn-

Banach and Riesz representation theorems. Given any distribution f ∈ H−1(�; K )

one defines the linear functional by Jf (Lϕ) = 〈 f, ϕ〉 for ϕ ∈ C∞0 (�). By the gen-

eralized Schwarz inequality (2.7) and the estimate (2.11), one has

|Jf (Lϕ)| ≤ C1‖ f ‖H−1(�;K )‖Lϕ‖L2(�), ϕ ∈ C∞0 (�),

and hence this functional is bounded on the subspace V of L2(�) of elements of

the form Lϕ with ϕ ∈ C∞0 (�). By the Hahn-Banach theorem, Jf extends to the

closure of V in L2(�) in a bounded way. Extension by zero on the orthogonal

complement of V gives a bounded linear functional on all of L2(�), and so, by the

Riesz representation theorem, there exists a u ∈ L2(�) satisfying (2.17). �

We note that the Sobolev space H 10 (�; K ) is a normal space of distributions

and that L is formally self-adjoint and hence the proof of Theorem 2.2 is classical

(cf. lemma B.1 of [29]). One can also show that the solution map from H−1(�; K )

to L2(�) is continuous with respect to the relevant norms.

Estimate (2.11) also shows that sufficiently strong solutions must be unique.

We say that u ∈ H 10 (�; K ) is a strong solution of the Dirichlet problem (2.1)–

(2.2) if there exists an approximating sequence un ∈ C20(�) such that

(2.18) ‖un − u‖H1(�;K ) → 0 and ‖Lun − f ‖L2(�) → 0 as n → +∞.

The following theorem is an immediate consequence of the definition:

THEOREM 2.3 Let � be any bounded region in R2 with piecewise C1 boundary.

Let K ∈ C1(R) be a type change function satisfying (1.3), (2.9), and (2.10). Then

any strong solution of the Dirichlet problem (2.1)–(2.2) must be unique.


We note that the class of admissible K is very large and includes the standard

models for transonic flow problems such as the Tricomi equation with K (y) = y

and the Tomatika-Tamada equation K (y) = A(1 − e−2By) with A, B > 0 con-

stants. Moreover, the result also holds for non-strictly-monotone functions such

as the Gellerstedt equation with K (y) = y|y|m−1 where m > 0. In this case,

one can check that in place of (2.14) it is enough to choose the dilation multiplier

introduced in [17]

(2.19) a ≡ 0, b = (m + 2)x, c = 2y.

We also note that no boundary geometry hypotheses have been made for the dis-

tributional existence result above; in particular, there are no starlike hypotheses on

the elliptic part and no subcharacteristic hypotheses on the hyperbolic part. These

kinds of hypotheses will, however, enter later on when we look for existence of

solutions in a stronger sense.

On the other hand, it is clear that the existence is in a very weak sense; too

weak, in fact, to be very useful. In particular, the sense in which the solution

vanishes at the boundary is only by duality, and one may not have uniqueness.

In fact, since L is formally self-adjoint, as noted above, the estimate (2.11) also

holds for LT = L , and this together with the Poincarè inequality (2.6) gives an

L2–L2 estimate for both L and LT for test functions u, v ∈ H 2(�) ∩ H 10 (�; K ).

These are the necessary and sufficient conditions of Berezanskiı̆ for a problem with

almost correct boundary condition (cf. chapter 2 of [3]). In order to be sure that the

distributional solution, which exists, is strong and satisfies the boundary condition

in the sense of traces, it would be enough to show that u ∈ H 2(�), but this will

not hold in general for f ∈ L2. In any case, the existence result is a first general

indication that while the closed Dirichlet problem is generically overdetermined

for regular solutions, it is generically not overdetermined if one looks for a solution

that is taken in a sufficiently weak sense. Moreover, while uniqueness generically

holds for strong solutions, one must show that such strong solutions exist.

EXAMPLE 2.4 Let � be any mixed domain containing the origin. Consider the

function

(2.20) u(x, y) = ψ E(x, y) =

{C−|9x2 + 4y3|5/6, (x, y) ∈ � ∩ D−(0),

0, (x, y) ∈ � \ D−(0),

where ψ(x, y) = 9x2 + 4y3, D−(0) = {(x, y) ∈ R2 : 9x2 + 4y3 < 0} is the

backward light cone from the origin, and

C− =3�(4/3)

22/3π1/2�(5/6).

Then u is a distributional solution in the sense (2.17) of the equation T u = yuxx +

uyy = f where

(2.21) f = −12yE

(7

2E + 3x Ex + 2yEy

).


The distribution E defined by (2.20) is the fundamental solution of Barros-Neto

and Gelfand [2] for the Tricomi operator with support in D−(0), and one verifies

that T u = f holds in the sense of distributions by repeating their calculations

with ψ E in the place of E . One has u ∈ L2(�) since u ∈ C0(�) and that f ∈

H−1(�; K ). The verification of (2.17) then follows. One notes that the restriction

of u does not vanish on all of the boundary and moreover that one should then be

able to build another distributional solution to the equation by adding a suitable

smooth function in the kernel of T .

As a final remark in this section we note that the approach of obtaining the a pri-

ori estimate (2.11) has its origins in the so-called a − b − c method of Friedrichs,

which was first used by Protter for showing uniqueness results for classical so-

lutions [26] to open boundary problems. The key difference is that the compact

support used here for the distributional existence must be replaced with a simple

condition of vanishing on the boundary, and hence boundary terms will remain in

the expression (2.13) whose signs must be compatible with the sign of the integral

over the domain. It is for this reason that technical hypotheses on the boundary

will enter.

3 Weak Solutions to the Dirichlet Problem

In this section, we show how to steer a course between the distributional exis-

tence and the strong uniqueness result for the Dirichlet problem by following the

path laid out by Didenko [5] for open boundary value problems. More precisely,

we will show that the Dirichlet problem is well-posed for weak solutions lying in

a suitable Hilbert space. Technical hypotheses on the type change function K and

the boundary will enter in the process.

To begin, we fix some notation and conventions that will be used throughout

this section. We will continue to assume that � is a bounded mixed domain with

piecewise C1 boundary with external normal field ν. Since the differential oper-

ator (2.1) is invariant with respect to translations in x , we may assume that the

origin is the point on the parabolic line AB := {(x, y) ∈ � : y = 0} with

maximal x-coordinate; that is, B = (0, 0). This will simplify certain formulas

without reducing the generality of the results. We will often require that � be

star-shaped with respect to the flow of a given (Lipschitz) continuous vector field

V = (V1(x, y), V2(x, y)); that is, for every (x0, y0) ∈ � one has Ft(x0, y0) ∈ �

for each t ∈ [0,+∞] where Ft(x0, y0) represents the time-t flow of (x0, y0) in the

direction of V . We recall that � is then simply connected and will have a V -star-

like boundary in the sense that V (x, y) · ν ≥ 0 for each regular point (x, y) ∈ ∂�

(cf. lemma 2.2 of [17]). We will continue to assume that the type change function

K belongs to C1(R) and satisfies (1.3) where additional hypotheses will be made

as needed.


We will also make use of suitably weighted versions of L2(�) and their prop-

erties. In particular, for K ∈ C1(R) satisfying (1.3) we define

L2(�; |K |−1) := { f ∈ L2(�) : |K |−1/2 f ∈ L2(�)},

equipped with its natural norm

‖ f ‖L2(�;|K |−1) =

[∫�

|K |−1 f 2 dx dy

]1/2

,

which is the dual space to the weighted space L2(�; |K |) defined as the equiva-

lence classes of square-integrable functions with respect to the measure |K |dx dy;

that is, with finite norm

‖ f ‖L2(�;|K |) =

[∫�

|K | f 2 dx dy

]1/2

.

One has the obvious chain of inclusions

(3.1) L2(�; |K |−1) ⊂ L2(�) ⊂ L2(�; |K |),

where the inclusion maps are continuous and injective (since K vanishes only on

the parabolic line, which has zero measure).

Under suitable hypotheses, we will show the existence of a unique H 10 (�; K )

solution to the Dirichlet problem (2.1)–(2.2) for each f ∈ L2(�; |K |−1). The

sense in which we will find a solution is contained in the following definition:

DEFINITION 3.1 We say that u ∈ H 10 (�; K ) is a weak solution of the Dirichlet

problem (2.1)–(2.2) if there exists a sequence un ∈ C∞0 (�) such that

(3.2) ‖un − u‖H10 (�;K ) → 0 and ‖Lun − f ‖H−1(�;K ) → 0 for n → +∞

or, equivalently,

(3.3) 〈Lu, ϕ〉 = −

∫�

(K uxϕx + uyϕy)dx dy = 〈 f, ϕ〉, ϕ ∈ H 10 (�, K ),

where 〈 · , · 〉 is the duality pairing between H 10 (�; K ) and H−1(�; K ), L is the

continuous extension defined in (2.8), and the relevant norms are defined in (2.5)

and (2.6).

The equivalence of (3.2) and (3.3) follows by integration by parts on C∞0 (�),

which is dense in H 10 (�; K ). We note that since L2(�; |K |−1) is a subspace of

L2(�) ⊂ H−1(�; K ), (3.2) and (3.3) make sense for f ∈ L2(�; |K |−1). Weak

solutions give an intermediate notion between the distributional solutions (2.17)

and the strong solutions (2.18) where the approximation property (3.2) is compat-

ible with the continuity property (2.8).


Our first result concerns the Gellerstedt operator; that is, with K of pure power

type

(3.4) K (y) = y|y|m−1, m > 0.

THEOREM 3.2 Let � be a bounded mixed domain with piecewise C1 boundary and

parabolic segment AB with B = 0. Let K be of pure power form (3.4). Assume

that � is star-shaped with respect to the vector field V = (−(m+2)x,−µy) where

µ = 2 for y > 0 and µ = 1 for y < 0. Then, for each f ∈ L2(�; |K |−1) there

exists a unique weak solution u ∈ H 10 (�; K ) in the sense of Definition 3.1 to the

Dirichlet problem (2.1)–(2.2).

Note that the restriction on the boundary geometry allows for both nonlenslike

and lenslike domains.

PROOF: For the existence, the basic idea is, as in the proof of existence for

distributional solutions (Theorem 2.2), to obtain an a priori estimate for setting up

a Hahn-Banach and Riesz representation argument. The needed estimate involves

one derivative less than in the distributional existence proof (cf. Lemma 2.1) and is

contained in the following lemma, which again applies also to LT = L:

LEMMA 3.3 Under the hypotheses of Theorem 3.2, one has the a priori estimate:

there exists C1 > 0 such that

(3.5) ‖u‖L2(�;|K |) ≤ C1‖Lu‖H−1(�;K ), u ∈ C∞0 (�).

We will call a domain � admissible for weak solutions to the Dirichlet problem

if the estimate (3.5) holds. Assuming that � is admissible, one defines a linear

functional Jf for ϕ ∈ C∞0 (�) by the formula Jf (Lϕ) = ( f, ϕ)L2(�), and the esti-

mate (3.5) together with the Cauchy-Schwarz inequality yields

|Jf (Lϕ)| ≤ ‖ f ‖L2(�;|K |−1) ‖ϕ‖L2(�;|K |)

≤ C1‖ f ‖L2(�;|K |−1)‖Lϕ‖H−1(�;K ), ϕ ∈ C∞0 (�).

Hence Jf is bounded on the subspace V of H−1(�; K ) of elements of the form Lϕ

with ϕ ∈ C∞0 (�). Now, as in the proof of Theorem 2.2, one obtains the existence

of u ∈ H 10 (�; K ) such that

(3.6) 〈u, Lϕ〉 = ( f, ϕ)L2(�), ϕ ∈ H 10 (�; K ),

where L is the self-adjoint extension defined in (2.8).

This distributional solution is a weak solution in the sense of Definition 3.1.

In fact, given a sequence un ∈ C∞0 (�) that approximates u in the norm (2.5),

the continuity property (2.8) shows that fn := Lun is norm convergent to some

element f̃ ∈ H−1(� : K ). One also has

(3.7) 〈un, Lϕ〉 = ( fn, ϕ)L2(�), ϕ ∈ H 10 (�; K ).

Taking the difference between (3.6) and (3.7) and passing to the limit shows that

f̃ = f and hence (3.2) holds.


For the uniqueness claim, again we use estimate (3.5). In fact, for fixed f , let

u, v ∈ H 10 (�; K ) be two weak solutions with approximating sequences {un} and

{vn} satisfying (3.2). Using the linearity of L and (3.5), one has that un − vn tends

to zero in L2(�; K ). Hence un − vn also tends to zero in L2(�) and in H 10 (�; K )

by the injectivity of the first inclusion in (3.1) and the Poincarè inequality (2.4).

Thus u = v in H 10 (�; K ), which finishes the proof of the theorem, modulo the

proof of the lemma. �

PROOF OF LEMMA 3.3: The basic idea is to estimate from above and below

the expression (I u, Lu)L2(�) for each u ∈ C∞0 (�) where v = I u is the solution to

the following auxiliary Cauchy problem:

(3.8)

{Mv := av + bvx + cvy = u in �

v = 0 on ∂� \ B

where B = (0, 0) is the right-hand endpoint of the parabolic line and

(3.9) a ≡ −1

4, (b, c) = −V = ((m + 2)x, µy).

We recall µ = 2 in �+ and µ = 1 in �−. The choice of the coefficients (a, b, c)

will ensure the positivity of a certain quadratic form used later in the estimate. We

note the difference with the proof of Lemma 2.1, in which an integral expression

v = I u replaces the differential expression Mu in (2.12). Since

(3.10) (I u, Lu)L2(�) = (v, Lu)L2(�) = (v, L Mv)L2(�),

one actually performs an a priori estimate for the third-order operator L M acting

on the auxiliary function v. We divide the rest of the proof into four steps.

Step 1. Existence and properties of the solution v to (3.8). We claim that for every

u ∈ C∞0 (�) there exists v ∈ C∞(�±)∩C0(�\ B) solving (3.8) with the additional

properties

lim(x,y)→B

v(x, y) = 0,(3.11) ∫�±

(|K |v2x + v2

y)dx dy < +∞.(3.12)

Hence, by defining v(B) = 0, one also has that

v ∈ C0(�) ∩ H 10 (�; K ).

To prove these claims, one first notes that since � is V-star-shaped, each flow line

of V , starting from the boundary and entering the interior, will stay in the closure

of � and that the only singular point of V is in the origin B. Hence, the method

of characteristics will give the existence of a unique v ∈ C∞(�±) ∩ C0(� \ B)

solving (3.8). We recall that the coefficients a and b are C∞, while c is globally

Lipschitz.


In order to verify the claims (3.11) and (3.12), we appeal to the explicit nature

of (b, c) = −V and the compact support of u. We parametrize the integral curves

of (b, c) by γ (t) = (x(t), y(t)) = (x0e(m+2)t , y0eµt) for (x0, y0) ∈ ∂� and t ∈

[−∞, 0]. We parametrize the boundary by �(s) where s ∈ [0, S] is the arc length

parameter, �(0) = �(S) = B, and we use the positive orientation on �. The

support supp(u) of u, which is compact and fixed, will be disjoint from each ε-

neighborhood of the boundary Nε(∂�) = {(x, y) ∈ � : dist((x, y), ∂�) ≤ ε} for

each ε sufficiently small. Moreover, there will be two critical values s (respectively,

s) of the arc length defined as the inf (respectively, sup) of the values of s for which

γ arrives at (x0, y0) = �(s) and γ intersects supp(u). In this way, the flow of −V

divides � into three regions �1, �2, and �3 corresponding to the intervals [0, s],

[s, s], and [s, S].

For each (x, y) ∈ �1 ∪ �3, the unique C1 solution v satisfying (3.8) must

vanish as u ≡ 0 there and v starts from zero. Similarly, for (x, y) in the wedge

Bε((0, 0)) ∩ �2, we can re-initialize the Cauchy problem at time t = T < 0 by

starting from points (xT , yT ) on the outer boundary of the wedge. The unique C1

solution along each flow line is

(3.13) ψ(t) = v(x(t), y(t)) = v(xT , yT )et/4, t ∈ (−∞, T ].

Since v is C1 in �, the variation in v(xT , yT ) is bounded for (xT , yT ) on the initial

data surface. The claims (3.11) and (3.12) are now easily verified.

Step 2. Estimate (3.10) from below. Since v ∈ C0(�) and Lu ∈ C∞0 (�), the

integrand is compactly supported and expression (3.10) is finite. Splitting along

the parabolic line (since v and c are only piecewise smooth)

(v, Lu)L2(�) =

∫�+

vLu dx dy +

∫�−

vLu dx dy

and applying the divergence theorem yields

(3.14)

(v, Lu)L2(�)

=1

2

∫�+∪�−

[αv2x + 2βvxvy + γ v2

y + v2La]dx dy

+1

2

∫∂�+∪∂�−

[2v(K ux , uy) − (Kv2x + v2

y)(b, c) − v2(K ax , ay)] · ν ds

where

α := −2K a − K bx + (K c)y =(2m + 1)K

2in �+,

α = −K

2in �−,

(3.15)

(3.16) β := −K cx − by = 0,


γ := −2a + bx − cy =2m + 1

2in �+,

γ =2m + 3

2in �−.

(3.17)

Using the smoothness and the compact support of u, the continuity of v, the fact

that a ≡ − 14, and (3.15)–(3.17), one finds that the identity (3.14) becomes

(v, Lu)L2(�) =2m + 1

4

∫�+

(|K |v2x + v2

y)dx dy

+1

4

∫�−

(|K |v2x + (2m + 2)v2

y)dx dy

−1

2

∫∂�+∪∂�−

(Kv2x + v2

y)(b, c) · ν ds.

(3.18)

The boundary integral in (3.18) is finite since the the quadratic form on the left is

finite, as noted at the beginning, and the double integrals on the right are finite by

(3.12). Moreover, the integrand in the boundary integral vanishes identically since

v vanishes identically in a sufficiently small neighborhood of each point on ∂�\ B,

and along the cut y = 0 we have (b, c) · ν ≡ 0. Hence, we have the estimate

(3.19) (v, Lu)L2(�) ≥1

4

∫�

(|K |u2x + u2

y)dx dy =1

4‖v‖2

H10 (�;K )

, u ∈ C∞0 (�).

Step 3. Estimate (3.10) from above. We use the generalized Schwarz inequality to

obtain

(3.20) (v, Lu)L2(�) ≤ ‖v‖H10 (�;K )‖Lu‖H−1(�;K ), u ∈ C∞

0 (�).

Step 4. Completing the estimate (3.5). Combining (3.19) and (3.20), one has

(3.21) ‖v‖H10 (�;K ) ≤ 4‖Lu‖H−1(�;K ), u ∈ C∞

0 (�).

Finally, it is easy to check that the first-order differential operator M is continuous

from H 10 (�; K ) into L2(�; |K |); that is, there exists CM > 0 such that

(3.22) ‖u‖L2(�;|K |) = ‖Mv‖L2(�;|K |) ≤ CM‖v‖H10 (�;K ), v ∈ H 1

0 (�; K ).

Combining (3.21) and (3.22) gives the desired estimate (3.5) with C1 = 4CM .

This completes Lemma 3.3 and hence Theorem 3.2 as well. �

Remark. We note that the multiplier (3.9) is almost the dilation multiplier (2.19);

it has been fudged a bit in the hyperbolic region in order to obtain the nonnegative

lower bound (3.19). In fact, one can slightly relax the hypotheses on � by choosing

µ to satisfy µ(m + 1) > m + 2 in �+ and µ(m + 1) < m + 2 in �−.


We complete the results of this section by showing how to eliminate almost

entirely the boundary geometry restrictions in the elliptic part of the domain and

indicate how to replace the type change functions K of pure power type with more

general forms. We denote by

�δ := {(x, y) ∈ � : y < δ}

in the following result, which shows that the Dirichlet problem for the Gellerstedt

equation is well-posed so long as the domain is suitably star-shaped for y < δ with

δ > 0 arbitrarily small.

THEOREM 3.4 Let �̂ be a bounded mixed domain that caps off an admissible

domain � in the sense that � ⊂ �̂, and there exists δ > 0 such that �̂δ = �δ.

Then �̂ is also admissible and the conclusion of Theorem 3.2 is also valid for �̂.

PROOF: The idea is originally due to Didenko [5] for open problems and has

been used in [16] for the Tricomi problem. For completeness, we sketch the main

points of the proof. We need to show that there exists C > 0 such that

(3.23) ‖u‖L2(�̂;|K |) ≤ C‖Lu‖H−1(�̂;K ), u ∈ C∞0 (�̂; K ).

For an arbitrary mixed domain �, one has the easy a priori estimate

(3.24) ‖u‖H10 (�+;K ) ≤ ‖Lu‖H−1(�+;K ), u ∈ C∞

0 (�+),

for the degenerate elliptic operator L on �+. In fact, one merely takes (a, b, c) =

(−1, 0, 0) and uses the argument of Section 2 (Lemma 2.1). Next, one picks a

cutoff function φ ∈ C∞(�̂) such that

φ(x, y) =

1, y ≥ 2δ/3,

χ(y), δ/3 ≤ y ≤ 2δ/3,

0, y ≤ δ/3,

where χ is any smooth transition function. One has that u = φu + (1 − φ)u with

φu ∈ C∞0 (�̂+) and (1 − φ)u ∈ C∞

0 (�). With �̂ in place of �, applying (3.24) to

the first term and (3.5) to the second term yields

(3.25) ‖u‖L2(�̂;|K |) ≤ ‖L(φu)‖H−1(�̂+;K ) + C1‖L((1 − φ)u)‖H−1(�;K ).

To complete estimate (3.23), one uses the Leibniz formula, the definition of the

norm in H−1(�̂+; K ), the generalized Schwarz inequality, and the continuity of

∂y : H 10 (�̂+; K ) → L2(�̂+) ⊂ L2(�̂+; |K |) to control the first term of (3.25),

and a similar argument to control the second term (see the proof of theorem 2.2 in

[16] for details). �

As a final improvement, one need not choose a type change function of pure

power type in order to have a well-posed problem. An example is contained in the

following proposition, in which the function µ = µ(y) defined by

(3.26) µ = (K/K ′)′


plays a key role. Both K and µ will be thought of as functions defined on � that

are independent of x .

PROPOSITION 3.5 Let K ∈ C2(R) be a type change function and � a bounded

mixed domain with B = (0, 0) such that K ′(y) > 0 in � and

(3.27) 2 sup�+

µ < 3 + 2 inf�+

µ,

where µ is defined by (3.26) and � is star-shaped with respect to the vector field

V = (−b0x,−2K/K ′) with

b0 =

{− 1

2+ ε + 2 sup�+ µ in �+

52

+ ε + 2 sup�− µ in �−

for some ε > 0. Then � is admissible and the Dirichlet problem is well-posed in

the sense of Theorem 3.2.

The proof follows precisely the argument used in the proof of Lemma 3.3,

where we note that in place of the multiplier (3.9) we now use a ≡ − 14

and

(b, c) = −V = (b0x, 2K/K ′). Note that the vector field V here merely gener-

alizes the one used in (3.9) for a pure power K with the associated anisotropic

dilation invariance. The technical hypothesis (3.27) is always satisfied in the pure

power case, but in general asks that �+ be thin in the y-direction if K has a large

variation. Of course, this thinness condition can be removed in the elliptic region

by applying Theorem 3.4. As a result, one obtains well-posedness for the Dirichlet

problem for the Tomatika-Tamada equation, for example, for any mixed domain

that has a suitably V -star-shaped hyperbolic part.

Finally, it should be noted that the norm H 1(�; K ) employed here has a weight

|K | that vanishes on the entire parabolic line and hence one might worry that the

solution is not locally H 1 due to the term |K |u2x . In fact, the solution does lie in

H 1loc(�) as follows from a microlocal analysis argument, if K is smooth enough,

using Hörmander’s theorem on the propagation of singularities (cf. [23]). On the

other hand, the norms used in [21] for treating the equation by way of a first-

order system were carefully constructed so as not to have weights vanishing on the

interior. Thus the method of [21] gives a direct proof of the lack of H 1 singularities

in the interior for the corresponding solution to the scalar problem in special cases.

We will return to the question of interior regularity in Section 5.

4 Well-Posedness for Mixed Boundary Conditions

Having treated the well-posedness for weak solutions to the Dirichlet problem,

we turn our attention to the other natural boundary condition for the transonic flow

applications, namely the conormal boundary condition. Here, we follow the path

laid out first by Morawetz [20] for open boundary value problems and then later

adapted to very special cases for closed boundary value problems of Dirichlet type

in [21] and conormal type in [25]. The basic idea is to rewrite the second-order


scalar equation for u as a first-order system for the gradient of u, which has the

effect of eliminating a derivative and allowing for a unified treatment of both nat-

ural boundary conditions. For general classes of type change functions and for

fairly general domains, we can obtain results for the Dirichlet problem (and hence

an alternative treatment to the method of Section 3) as well as for mixed bound-

ary conditions. For now, we are unable to make the technique work directly on

the full conormal problem and hence the result of Pilant [25] (in the special case

K (y) = sgn(y)) remains the only result on the full conormal problem.

As a first step, we recall why boundary conditions of Dirichlet and conormal

type are natural for the class of equations (2.1). As already noted, the operator L

is formally self-adjoint, and one has the following Green’s identity for sufficiently

regular functions u, v:∫�

(uLv − vLu)dx dy =

∫∂�

(vuν − uvν)ds

where uν = (K ux , uy) · ν is the so-called conormal derivative with respect to

the divergence form operator L = div ◦(K ∂x , ∂y). In this way, one expects that

representation formulas for solutions to (2.1)–(2.2) might be available using as

data the forcing term f in � as well as the values of u and uν on the boundary, as

happens for the the Laplace equation in terms of the Dirichlet and Neumann data.

This approach has been used for open boundary value problems with some success

(cf. [10, 28] and the references therein).

We will continue to use much of the notation and conventions that were intro-

duced in Section 3. In particular, � is a bounded domain with piecewise C1 bound-

ary; the parabolic line AB = {(x, y) ∈ � : y = 0} has its maximal x-coordinate

in B = (0, 0); the boundary ∂� will often be star-shaped with respect to a given

(Lipschitz-) continuous vector field V ; the type change function K ∈ C1(R) satis-

fies (1.3) and additional conditions as needed.

Before giving the main results, we briefly recall the formulation of the boundary

value problems via first-order systems as done in [21, 25]. In place of the second-

order differential equation for the scalar-valued u

(4.1) Lu = K uxx + uyy = f,

with boundary conditions of Dirichlet

(4.2) u = 0 on � ⊆ ∂�

or conormal type

(4.3) uν = 0 on ∂� \ �,

one considers the system Lv = g

(4.4) Lv =

[K ∂x ∂y

∂y −∂x

] [v1

v2

]=

[g1

g2

].


In the case where (g1, g2) = ( f, 0) and v = ∇u, then a solution v to (4.4)

gives rise to a solution u of (4.1). In order to give a suitable formulation of weak

solutions to (4.4) and in order to build in the analogous boundary conditions to

(4.2)–(4.3), one exploits the following integration-by-parts formula, which is valid

for every pair of sufficiently regular functions v, ϕ on �:

(4.5) (Lv, ϕ) = −(v,Lϕ) +

∫∂�

ϕ1(Kv1 dy − v2 dx) −

∫∂�

ϕ2(v1 dx + v2 dy),

where in this section, we will denote by

(4.6) (v, ϕ) =

∫�

(v1ϕ1 + v2ϕ2)dx dy

the scalar product on H(�) = L2(�; R2). Keeping in mind that v should be

thought of as ∇u, the Dirichlet condition (4.2) for u becomes the condition

(4.7) v1 dx + v2 dy = 0 on � ⊆ ∂�

on v, which of course corresponds to u identically constant on �. As for the conor-

mal condition (4.3), one places the following condition on v:

(4.8) Kv1 dy − v2 dx = 0 on ∂� \ �.

We will make use of several spaces of vector-valued functions analogous to those

used in the solvability discussions of Sections 2 and 3. We will find solutions in

the Hilbert space HK(�) of all measurable functions v = (v1, v2) on � for which

the norm

(4.9) ‖v‖HK=

[∫�

(|K |v21 + v2

2)dx dy

]1/2

is finite, where we denote the scalar product by

(4.10) (v, ϕ)K =

∫�

(|K |v1ϕ1 + v2ϕ2)dx dy.

Using the inner product (4.6), the dual space to HK(�) can be identified with

the space HK−1(�) in which |K |−1 replaces |K | in (4.9) and (4.10). These spaces

are special cases of the Hilbert space HA(�) = {v : Av ∈ H(�)} where A is

a piecewise continuous matrix-valued function on �, which is often assumed to

be invertible almost everywhere. In particular, the spaces HK(�),HK−1(�) are of

this general form with the diagonal matrix K having entries |K |±1 and 1. We will

denote by A∗ the transpose of the matrix A.

Using the formulas (4.5), (4.7), and (4.8), we have the following notion of

weak solutions:


DEFINITION 4.1 Let � be a bounded mixed domain and g such that K−1M∗g ∈

H(�) for a fixed matrix-valued function M that is piecewise continuous and in-

vertible almost everywhere. Let � ⊆ ∂� be a possibly empty subset. A weak

solution to the system (4.4) with the Dirichlet condition on � ⊆ ∂� and conormal

condition on ∂� \ � is the function v ∈ HK (�) such that

−(v,Lϕ) = (g, ϕ),

for each test function ϕ such that

(4.11) ϕ ∈ C1(�; R2) and K

−1Lϕ ∈ H(�)

and

ϕ1 = 0 on �,(4.12)

ϕ2 = 0 on ∂� \ �.(4.13)

A weak existence theorem for the system follows directly from a suitable a pri-

ori estimate. We record the following general result, which has its origins in [20]

for problems with open boundary conditions.

LEMMA 4.2 Let � be a bounded mixed domain and � a potentially empty subset of

∂�. Assume that there exists a matrix-valued function M for which the following

a priori estimate holds: there exists C > 0 such that

(4.14) ‖KM−1ϕ‖H(�) ≤ C‖K−1

Lϕ‖H(�)

holds for each test function satisfying (4.11)–(4.13). Then there exists a weak

solution u in the sense of Definition 4.1.

PROOF: The proof is classical (for example, see the proof of theorem 3 in [21]

or section 2 of [25]). For completeness, we give a sketch. One defines the linear

functional Jf (Lϕ) = (g, ϕ) that by the Cauchy-Schwarz inequality and the a priori

estimate (4.14) yields

|Jf (Lϕ)| = |(K−1M

∗g,KM−1ϕ)|

≤ ‖K−1M

∗g‖H(�)‖KM−1ϕ‖H(�)

≤ CgC‖K−1Lϕ‖H(�),

where we note K is diagonal and hence self-adjoint and Cg is a constant that de-

pends on the g, which is fixed. Hence Jf gives a bounded linear functional on

the subspace W of HK−1(�) of elements of the form Lϕ. One then extends Jf to

the whole of HK−1(�) in the standard way (first by continuity on W and then by

zero on its orthogonal complement). The Riesz representation theorem then gives

w ∈ HK−1(�) such that

(g, ϕ) = (w,Lϕ)K−1 = (K−2w,Lϕ)

and hence v = −K−2w is the desired solution. �


Lemma 4.2 shows that for a given domain � and a given type change function

K , it is then enough to find a suitable multiplier matrix M for which the estimate

(4.14) holds. One again, a suitable version of the a − b − c method as developed

in [20] is the key.

LEMMA 4.3 Let � be a bounded mixed domain and � a potentially empty subset

of ∂�. Let K ∈ C1(R) be a type change function satisfying (1.3). Assume that

there exists a pair (b, c) of continuous and piecewise C1 functions on � such that

(4.15) b2 + K c2 > 0 on �

and there exists a constant δ > 0 such that for each (x, y) ∈ � and each � ∈ R2

one has

(4.16) Q(�) := α(x, y)�21 + 2β(x, y)�1�2 +γ (x, y)�2

2 ≥ δ(|K (y)|�21 +�2

2),

where

(4.17) α = K bx − (K c)y, β = K cx + by, γ = cy − bx .

Assume also that

b dy − c dx ≤ 0 on �,(4.18)

K (y)(b dy − c dx) ≥ 0 on ∂� \ �.(4.19)

Then there exists a matrix M for which the a priori estimate (4.14) holds for each

test function satisfying (4.11)–(4.13).

Before giving the proof, we make a few remarks. Lemma 4.3 together with

Lemma 4.2 reduces the weak existence to the art of finding a suitable multiplier

pair (b, c). Notice also that in this system approach the multiplier a plays no role

since one is working on v = ∇u in the place of u. The conditions (4.18) and (4.19)

say that � and ∂� are suitably starlike with respect to the vector field V = (b, c).

In terms of the flow of V one is asking for inflow in the case of Dirichlet conditions

and inflow/outflow in the case of conormal conditions depending on whether y is

negative/positive. This need for reversing orientation of the flow across y = 0

makes the problem of finding a multiplier for the conormal problem particularly

difficult. The proof of Lemma 4.3 is essentially classical, as done in [21, 25].

The minor difference here is the possible mixture of boundary conditions and the

a priori choice of the norm in which to find solutions, as governed by the right-hand

side of (4.16). More generally, one could replace (4.16) by asking only that Q > 0

on � and use this quadratic form to define a norm in a different way.

PROOF OF LEMMA 4.3: Given � and (b, c) satisfying (4.15)–(4.19), one de-

fines a multiplier matrix M by the recipe of [20]:

M =

[b c

−K c b

],


which is invertible except possibly at B = (0, 0) by (4.15). Given any test function

ϕ satisfying (4.11)–(4.13), one defines

� = M−1ϕ or ϕ = M�

and tries to estimate (Lϕ,M−1ϕ) = (LM�,�) from above and below. Notice

that the conditions (4.12) and (4.13) on ϕ become

(M�)1 := b�1 + c�2 = 0 on �,(4.20)

(M�)2 := −K c�1 + b�2 = 0 on ∂� \ �.(4.21)

One easily checks that by integrating by parts as in (4.5) and using (4.20) and

(4.21) one has

2(LM�,�) =

∫�

Q(�) dxdy +

∫�

−1

c2�2

1(b2 + K c2)(b dy − c dx)

+

∫∂�\�

1

b2�2

1(b2 + K c2)K (y)(b dy − c dx)

≥

∫�

Q(�)dx dy,

where we have used (4.15), (4.18), and (4.19) and Q is defined in (4.16). Using

the lower bound in (4.16) one has

(4.22) 2(LM�,�) ≥ δ

∫�

(|K |�21 + �2

2)dx dy = δ‖KM−1ϕ‖2

H(�).

Then estimating from above, one has

2(LM�,�) = 2(K−1Lϕ,KM

−1ϕ)

≤ 2‖K−1Lϕ‖H(�)‖KM

−1ϕ‖H(�),(4.23)

by the Cauchy-Schwarz inequality since ϕ satisfies (4.11). Combining (4.22) and

(4.23) yields

‖KM−1ϕ‖H(�) ≤

2

δ‖K−1

Lϕ‖H(�),

as claimed. �

Our first existence result for the system (4.4) with Dirichlet or mixed boundary

conditions is the following analogue of Theorem 3.2 for type change functions of

pure power type K .

THEOREM 4.4 Let K = y|y|m−1 with m > 0 be a pure power type change function.

Let � be a mixed domain, and let � be either the entire boundary ∂� or the elliptic

boundary (∂�)+. Assume that (4.15), (4.18), and (4.19) hold with respect to the

multiplier pair

(4.24) (b, c) = (−(m + 2)x),−µy)


where µ = 2 in �+ and µ = 1 in �−. Then, for each g such that K−1M∗g ∈

H(�), there exists a weak solution v ∈ HK(�) in the sense of Definition 4.1 to the

system (4.5) with Dirichlet conditions on � and conormal conditions on ∂� \ �.

PROOF: By Lemmas 4.2 and 4.3 one needs only to check that the multiplier

pair (4.24) gives rise to the inequality (4.19) for some δ > 0. One easily checks

that δ = min{1, m} will do. �

Using the same proof, one can generalize Theorem 4.4 by replacing the pure

power type change function K by an almost pure power K that satisfies the hy-

potheses of Proposition 3.5. Choosing again (b, c) = (−b0x,−2K/K ′) as the

multiplier pair defines a class of mixed domains � for which one has a weak solu-

tion to the system for the Dirichlet or the mixed boundary value problem.

We conclude with a few additional remarks. In order to get uniqueness of the

weak solutions, one needs to show that there is sufficient regularity. This can be

accomplished by following the techniques laid out in [21, 25], which in turn use

ideas from [9, 15] in which mollifying in the x-direction plays a key role. In a

similar way, if there is enough regularity, then in the case (g1, g2) = ( f, 0) one can

use the second equation in system (4.4) in order to define a potential function u for

which v = ∇u, and this u will solve the scalar equation (4.1) with the associated

boundary conditions (4.2)–(4.3).

5 Local Regularity Results

In this section we investigate the local regularity of the weak solution to the

Dirichlet problem for the Tricomi equation. That is, for the special case K (y) = y,

we consider the regularity of a solution u ∈ H 10 (�; K ) to the problem

T u = yuxx + uyy = f in �,(5.1)

u = 0 on ∂�,(5.2)

where � is an admissible mixed domain for the existence result of Theorem 3.4

in the case of the Tricomi equation. For example, it suffices to take � whose

hyperbolic region and the part of the elliptic region with y > 0 small is suitably

star-shaped. When it is necessary to ensure the existence of a unique solution, we

will assume that f ∈ L2(�; |y|−1) in accordance with Theorems 3.2 and 3.4. We

recall that there then exists a unique solution u ∈ H 10 (�; y) where the weighted

norm defined by (2.7) in this special case becomes

(5.3) ‖u‖H10 (�;y) =

[∫�

(|y|u2x + u2

y)dx dy

]1/2

.

For the applications to transonic flow, it would be important to know that the

solutions were at least continuous in the interior. We will show that this is so if

f is slightly more regular than what is required for the existence and uniqueness.


We will show that u is locally in H 2 and hence continuous by the Sobolev embed-

ding theorem. Our proof will require knowing that the solution is already locally

H 1 using only the weaker assumption that f belongs to L2(�; |y|−1), which we

have already noted follows from Hörmander’s theorem on the propagation of sin-

gularities. We will give instead a direct argument for the local H 1 regularity that

avoids any fancy machinery; this argument was suggested to us by Louis Niren-

berg. His argument involves a formal estimate which states that locally one can

remove the presence of the weight |y| in the norm (5.3) for sufficiently smooth so-

lutions, where mollifying in the x-direction yields the needed regularity to justify

the estimate.

We begin with the formal estimate.

LEMMA 5.1 (Nirenberg) Let u be a weak solution to the inhomogeneous Tricomi

equation (5.1). Assume that

F :=

∫�

f 2 dx dy < +∞,

E :=

∫�

(|y|u2x + u2

y)dx dy < +∞.

If u is smooth enough, then for each compact subdomain G ⊂ � there exists a

constant C = C(�, G) such that∫G

u2x dx dy ≤ C(E + F).

PROOF: Pick a smooth cutoff function ζ ∈ C∞0 (�) such that 0 ≤ ζ ≤ 1 and

ζ ≡ 1 on G. One merely computes by a sequence of integrations by parts, which

will need to be justified later. Starting from∫�

ζu2x dx dy =

∫�

[(yζ )yu2x − yζyu2

x ] dx dy,

we find ∫�

ζu2x dx dy = −

∫�

yζyu2x dx dy − 2

∫�

yζux uxy dx dy

≤ C1 E + 2

∫�

(yζux)x uy dx dy

≤ C2 E + 2

∫�

yζuxx uy dx dy.


Now we use equation (5.1) to say that∫�

ζu2x dx dy ≤ C2 E + 2

∫�

ζ( f − uyy)uy dx dy

≤ C3(E + F) − 2

∫�

ζuyyuy dx dy,

and then additional integration by parts finishes the claim:∫�

ζu2x dx dy ≤ C3(E + F) −

∫�

ζ(u2y)y dx dy

≤ C(E + F).

�

Remark. As for the needed regularity, observe that one needs to know that u2x ∈

L1loc(�) to start the estimate; then u ∈ H 2

loc(�) would be enough to justify the

various integration by parts performed.

We now show that the weak solution to (5.1)–(5.2), when mollified in the x-

direction, is regular enough to justify the formal estimate and to pass to a limit.

To this end, we fix a smooth bump function of one variable η ∈ C∞0 (R) such that

0 ≤ η ≤ 1 and∫

Rη(t)dt = 1 and define, for ε > 0, ηε(t) = ε−1η(t/ε) so that

ηε ∈ C∞0 (B(0; ε)) and

∫R

ηε(t)dt = 1. Given any u ∈ L1loc(�), measurable and

locally integrable with respect to Lebesgue measure, one defines its mollification

in x in the usual way,

uε(x, y) :=

∫R

ηε(x − t)u(t, y)dt

=

∫B(0;ε)

ηε(t)u(x − t, y)dt,

(5.4)

where we extend u by 0 outside of �. By Fubini’s theorem, uε( · , y) is well-

defined for every ε > 0 and for almost every y ∈ π2(�), where π2 is the pro-

jection onto the second coordinate of �, which we assume to be bounded, open,

and simply connected. For the exceptional values of y one can take uε to be 0.

The following lemma summarizes known properties of the mollification in one

variable, whose proof is classical. In particular, properties (i) through (iv) along

slices with y constant are well-known (cf. appendix C of [6], for example) and

properties (v) through (vii) follow by standard analysis (cf. [15]), where we define

I (y) = {x ∈ R : (x, y) ∈ �} and Iε(y) = {x ∈ I (y) : x ± ε ∈ I (y)}.


LEMMA 5.2 Let u ∈ L1loc(�). Then one has for almost every y ∈ π2(�):

(i) uε( · , y) ∈ C∞(Iε(y)) and for each k ∈ N

Dkx uε(x, y) =

∫I (y)

Dkηε(x − t)u(x, t)dt ∀x ∈ Iε(y).

(ii) uε( · , y) → u( · , y) almost everywhere on I (y) as ε → 0.

(iii) If u( · , y) ∈ C0(I (y)), then uε( · , y) → u( · , y) uniformly on compact

subsets of I (y).

(iv) If 1 ≤ p < ∞ and u ∈ Lp

loc(I (y)), then uε( · , y) → u( · , y) in Lp

loc(I (y)).

Moreover, let u ∈ L2(�). Then one has:

(v) uε ∈ L2(�) and uε → u in L2(�) for ε → 0, where uε is the mollification

in x of u, defined by (5.4) for almost every y ∈ π2(�) and 0 otherwise.

(vi) Dkx Jε : L2(I (y)) → L2(I (y)) is bounded for every k ∈ N, where Jεu :=

uε .

(vii) Dkx Jε : L2(�) → L2(�) is bounded for every k ∈ N.

Up to now, we have recalled general properties of mollifying in x for arbitrary

locally integrable and square-integrable functions. One can say more for u a weak

solution to (5.1), an equation whose coefficients do not depend on x .

LEMMA 5.3 Let u ∈ H 1(�; y) be a weak solution to (5.1) with f ∈ L2(�). Then

uε ∈ H 2(�) is a weak solution to the equation

(5.5) T uε = fε

where fε = Jε f is the mollification in x of f .

PROOF: Since we assume that u ∈ H 1(�; y), we have that u, |y|1/2ux , and

uy ∈ L2(�), and by Lemma 5.2 we have uε ∈ L2(�) and Dkx uε ∈ L2(�) for

each k ≥ 1. As for Dyuε , recall that one says that the weak y-derivative of uε ∈

L2(�) ⊂ L1loc(�) exists if there exists vε ∈ L1

loc(�) such that

(5.6)

∫�

uε Dyϕ dx dy = −

∫�

vεϕ dx dy ∀ϕ ∈ C∞0 (�).

The obvious candidate is vε := Jε Dyu, which is well-defined and belongs to L2(�)

by Lemma 5.2. Standard calculations show that (5.6) holds with vε := Jε Dyu, and

so Dyuε exists, belongs to L2(�), and equals (Dyu)ε .

Next we notice the important fact that the mollification in x commutes with

multiplication by y (or any function of y for that matter), and one thus has the


following commutator identities: for every g ∈ L2(�) and for every ϕ ∈ C∞0 (�),

one has

(yg)ε = ygε,(5.7)

Dx(ϕε) = (Dxϕ)ε, Dy(ϕε) = (Dyϕ)ε,(5.8) ∫�

gϕε dx dy =

∫�

gεϕ dx dy.(5.9)

Next we claim that if u ∈ L2(�) solves (5.1) in the sense of distributions, then

uε solves (5.5) in the sense of distributions. Indeed, starting from

(5.10)

∫�

u(yϕxx + ϕyy)dx dy =

∫�

f ϕ dx dy, ϕ ∈ C∞0 (�),

take ε > 0 small enough so that the support of ϕε is inside �, apply (5.10) to ϕε ,

and use properties (5.8) and (5.7) to find∫�

u((yϕxx)ε + (ϕyy)ε)dx dy =

∫�

f ϕε dx dy, ϕ ∈ C∞0 (�).

Then use formula (5.9) to find

(5.11)

∫�

uε(yϕxx + ϕyy)dx dy =

∫�

fεϕ dx dy, ϕ ∈ C∞0 (�),

which gives the claim.

By rewriting (5.11) in the form∫�

uεϕyy dx dy =

∫�

( fε − y(uε)xx)ϕ dx dy, ϕ ∈ C∞0 (�),

where we know fε − y(uε)xx ∈ L2(�), we see that the weak derivative D2yuε

exists and belongs to L2(�). Then since Dkyuε ∈ L2(�), we have about any point

(x0, y0) ∈ � there is a closed rectangle Qδ = Iδ × Jδ ⊂ � on which uε ∈

H 2(Iδ, L2(Jδ)), which embeds into C1(Jδ, L2(Iδ)). That is, after redefining uε on

a set of measure 0 it is a C1-function of y with values that are square-integrable

in x .

Finally, we claim that the weak derivative Dx Dyuε exists and belongs to L2(�).

This is because Dx Dy Jεu = Dx(Jε Dyu), which lies in L2(�) since Dyu ∈ L2(�).

Hence we have uε ∈ H 2(�), which completes the proof. �


We now have the needed preliminaries in order to prove the local regularity

results.

THEOREM 5.4 Let � be a mixed domain and f ∈ L2(�). If u ∈ H 1(�, y) is a

weak solution of the inhomogeneous Tricomi equation T u = f , then u ∈ H 1loc(�).

PROOF: Let uε be the mollification of u with respect to x that belongs to H 2(�)

and solves T uε = fε by Lemma 5.3. Applying Lemma 5.1 to uε yields the

inequality

(5.12)

∫�

ζ(uε)2x dx dy ≤ C

∫�

(|y|(uε)2x + (uε)

2y + f 2

ε )dx dy,

where ζ is a cutoff as in Lemma 5.1. By Lemma 5.2(v), one has that the right-

hand side of (5.12) converges to the same expression with u in place of uε , while

the integrand on the left-hand side converges almost everywhere to ζu2x by Lemma

5.2(ii). Applying Fatou’s lemma to (5.12) then gives the needed estimate∫�

ζu2x dx dy ≤ C

∫�

(|y|u2x + u2

y + f 2)dx dy.

and the theorem follows. �

Finally, higher-order local regularity can be achieved by assuming more regu-

larity on the forcing term f .

THEOREM 5.5 Let u ∈ H 10 (�, y) be the unique weak solution to the Dirichlet

problem for the inhomogeneous Tricomi equation T u = f with f ∈ L2(�, |y|−1).

If, in addition, fx ∈ L2(�, |y|−1), then u ∈ H 2loc(�).

PROOF: The weak derivative ux is well-defined as an element of L2(�, |y|)

and belongs to L2loc(�) by Theorem 5.4. Moreover, it is a weak solution to the

equation T ux = fx in the sense of distributions, although no claim is made about

its boundary values. One could call this an interior weak solution in analogy with

the definition of Sarason [27] for first-order systems. On the other hand, since

fx ∈ L2(�, |y|−1) there also exists a unique weak solution w ∈ H 10 (�; y) to the

Dirichlet problem for the equation T w = fx . Such a solution is also an interior

weak solution of the same equation and hence we have the identity

(w, T v)L2(�) = ( fx , v)L2(�) = (ux , T v)L2(�) ∀v ∈ C∞0 (�).

Now, if one knew that for each compact subdomain G of � one has

(5.13) {g ∈ L2(G) : g = T v, v ∈ C∞0 (�)} is dense in L2(G),

then the result would follow. Indeed, by Theorem 5.4 we have w ∈ H 1(G) and so

ux agrees almost everywhere on G with a function in H 1(G), and hence uxx , uxy ∈


L2(G) for each compact G. By using the equation T u = f , it then follows that

uyy ∈ L2(G) and hence u ∈ H 2loc(�).

It remains to verify the density claim (5.13). For each g ∈ L2(G), extend it

to all of � by defining it to be zero outside G and call the resulting extension g.

By Theorem 2.2 there exists an interior weak solution U ∈ L2(�) to the equation

T U = g ∈ L2(�). We claim that U is also an interior strong solution in the sense

that there exists a sequence of functions Un ∈ C∞0 (�) such that

(5.14) limn→+∞

[‖Uu − U‖L2(G) + ‖T Un − g‖L2(G)] = 0.

For the case of first-order systems, this was proved by Friedrichs [8] via mollifica-

tion (see also [27]). For second-order scalar equations, it is not always true that the

result follows by simple mollification; there exist counterexamples due to Hörman-

der [11]. However, if the differential operator T is principally normal and � admits

a function ψ ∈ C2(�) such that the level sets of ψ are pseudoconvex with respect

to T , then one has the approximation property (5.14) for the transpose T T by a du-

ality argument and suitable a priori estimates as shown by Hörmander (cf. theorem

8.7.4 of [12]). In our case, T has real coefficients and so T is principally nor-

mal. Moreover, T T = T is formally self-adjoint and the function ψ(x, y) = −y

has the desired pseudoconvexity property with respect to T as pointed out by Hör-

mander in section 8.6 of [12]. This justifies (5.14) and completes the proof of the

theorem. �

Remark. One can generalize Theorems 5.4 and 5.5 to include more general type

change functions K (y) provided that K is sufficiently smooth in order to apply

Hörmander’s results on singularity propagation as well as the weak equals strong

result used to obtain the density (5.13).

Acknowledgment. The first and third authors wish to thank the Courant Insti-

tute for its hospitality and stimulating environment during the period in which this

work was begun.

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DANIELA LUPO

Politecnico di Milano

Dipartimento di Matematica

“F. Brioschi”

Piazza Leonardo da Vinci, 32

20133-MILANO

ITALY

E-mail: daniela.lupo@

mate.polimi.it

CATHLEEN S. MORAWETZ

Courant Institute

251 Mercer Street

New York, NY 10012

E-mail: [email protected]

KEVIN R. PAYNE

Università di Milano

Dipartimento di Matematica

“F. Enriques”

Via Saldini, 50

20133-MILANO

ITALY

E-mail: [email protected]

Received September 2005.

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