Hardy inequalities and asymptotic behavior for heat kernels

Outline

Hardy inequalities and asymptotic behavior forheat kernels

Enrique Zuazua

Basque Center for Applied Mathematics (BCAM)Bilbao, Basque Country, Spain

[email protected]

http://www.bcamath.org/zuazua/

Analysis on Graphs and its Applications Follow-up Meeting,Cambridge, July 2010

Enrique Zuazua Hardy inequalities and asymptotic behavior for heat kernels

Outline

Outline


Outline

Outline

1 Justificacion

2 Introduction and motivation

3 The Hardy inequality

4 The Dirichlet problem for the parabolic operator

5 The Cauchy problem

6 Boundary singularities

7 Twisted domains

8 Concluding remarks and open problems


Justificacion Introduction The Hardy inequality The Dirichlet problem for the parabolic operator The Cauchy problem Boundary singularities Twisted domains Concluding remarks and open problems







Mutual anihilation:



Mutual anihilation:



Mutual anihilation:



Mutual anihilation:



Mutual anihilation:



Mutual anihilation:

See the lecture by V. Chernyshev for similar dynamics for theSchrodinger equation on graphs.



Kannai transform allows to transform the wave equation into theheat equation (Y. Kannai, 1977)

et∆ϕ =1√4πt

∫ +∞

−∞e−s2/4tW (s)ds

where W (x , s) solves the corresponding wave equation with data(ϕ, 0).

Wss −∆W = 0 + Kt − Kss = 0 → Ut −∆U = 0,

Wss −∆W = 0 + iKt − Kss = 0 → iUt −∆U = 0.

This can be actually applied in a more general abstract context(Ut + AU = 0) but not when the equation has time-dependentcoefficients.





Motivation & Goal

Motivation:

PDE with singular potentials arising in combustion theory andquantum mechanics.

Goal:

Revise the existing theory of well-posedness, asymptoticbehavior, control, etc. when replacing −∆ by −∆− λ

|x |2 both

in the elliptic and in the evolution context.



Part of the literature on singular elliptic and parabolic problems:

S. Chandrasekhar, An introduction to the study of stellarstructure, New York, Dover, 1957.

I. M. Gelfand, Some problems in the theory of quasilinearequations, Amer. Math. Soc. Transl., 29 (1963), 295-381.

J. Serrin, Pathological solution of an elliptic differentialequation, Ann. Scuola Norm. Sup. Pisa, 17 (1964), 385–387.

D. S. Joseph & T. S. Lundgren, Quasilinear Dirichletproblems driven by positive sources, Arch. Rat. Mech. Anal.,49 (1973), 241-269.

F. Mignot, F. Murat, J.-P. Puel, Variation d’un point deretournement par rapport au domaine, Comm. P. D. E. 4(1979), 1263-1297.

P. Baras, J. Goldstein, The heat equation with a singularpotential, Trans. Amer. Math. Soc. 284 (1984), 121–139.......



Examples:

Example 1:−∆u − µ(1 + u)p = 0,

p > n/(n − 2), µ =2

p − 1(n − 2p

p − 1).

u(x) = |x |−2/(p−1) − 1

After “linearization”:

−∆v − λ

|x |2v = f .

with

λ =2p

p − 1(n − 2p

p − 1).



Example 2:−∆u − λeu = 0, λ = 2(N − 2)

u(x) = −2log(|x |).

After “linearization”:

−∆v − λ

|x |2v = f .



Example 3:

−∆u = |∇u|q, u(x) = cq(|x |−(2−q)/(q−1) − 1)

Linearization:

−∆v = q|∇u|q−2∇u · ∇v ∼ −∆v = µ1

|x |2x · ∇v .

This type of singular problem, with singularities in the first orderterm, can be treated similarly as the previous ones. This is seeneasily when analyzing its coercivity since∫

1

|x |2x · ∇v vdx =

1

2

∫1

|x |2x · ∇(v 2)dx =

N − 2

2

∫v 2

|x |2dx .



D. Joseph et al., 1973.Enrique Zuazua Hardy inequalities and asymptotic behavior for heat kernels


The Cauchy problem

ut −∆u − λ

|x |2 u = 0 in Q

u = 0 on Σu(x , 0) = u0(x) in Ω.

Baras-Goldstein (1984), N ≥ 3:

Global existence for λ ≤ λ∗ = (N − 2)2/4;

Instantaneous blow-up if λ > λ∗ = (N − 2)2/4.

Explanation: Hardy’s inequality:

λ∗

∫Ω

ϕ2

|x |2dx ≤

∫Ω|∇ϕ|2dx .

Optimal not achieved constant: ϕ = |x |−(N−2)/2.

Warning! In dimension N = 2 this inequality fails....λ∗ = 0.1

1See Adimurthi & Sandeep for a logarithmic correction in dimension N = 2.Enrique Zuazua Hardy inequalities and asymptotic behavior for heat kernels


Preliminaries on Hardy inequalities:

The classical Hardy inequality ensures that

(N − 2)2

4

∫RN

ϕ2

|x |2dx ≤

∫RN

|∇ϕ|2dx .

The proof is easy:

ϕ(x)

|x |= −

∫ ∞1

x

|x |· ∇ϕ(tx)dt.

And apply the Minkowski inequality in L2(RN).

Of course, it also holds in H10 (Ω) for any domain Ω.

It guarantees the coercivity of the operator −∆− λ/|x |2 inH1

0 (Ω), for λ < λ∗ = (N − 2)2/4.



Hardy-Poincare inequality

But this inequality fails to yield coercivity for the critical valueλ∗ = (N − 2)2/4.H. Brezis-J. L. Vazquez, 1997:

λ∗

∫Ω

ϕ2

|x |2dx + C (Ω)

∫Ωϕ2dx ≤

∫Ω|∇ϕ|2dx , ∀ϕ ∈ H1

0 (Ω).

Later improved 2: 0 < s < 1,

λ∗

∫Ω

ϕ2

|x |2dx + C (Ω)||ϕ||2s ≤

∫Ω|∇ϕ|2dx , ∀ϕ ∈ H1

0 (Ω).

−∆− λ∗|x |2 I is almost as coercive as −∆.

2J. L. Vazquez & E. Z. The Hardy inequality and the asymptotic behaviorof the heat equation with an inverse square potential. J. Funct. Anal., 173(2000), 103–153; Adimurthi, N. Chaudhuri and M. Ramaswamy. An improvedHardy-Sobolev inequality and its applications. Proc. AMS 130 (2002),489-505; Adimurthi and K. Sandeep, Existence and non-existence of the firsteigenvalue of the perturbed Hardy-Sobolev operator, Proceedings of the RoyalSociety of Edinburgh, 132A, 1021-1043, 2002.



For the critical value λ∗, the elliptic operator −∆− λ∗|x |2 I plays the

role of −∆ but in the slightly larger space H(Ω), the closure ofD(Ω) with respect to the norm

||ϕ||H =[ ∫

Ω

[|∇ϕ|2 − λ∗

∫Ω

ϕ2

|x |2]dx]1/2

.

The elliptic and parabolic/hyperbolic theories (existence,uniqueness, and to a large extent control) are then the same byreplacing H1

0 (Ω) by H(Ω).

But note that this only happens in the bounded domain case sincethe Poincare remainder term can not catched up in the wholespace RN .



Idea of the proof:

Ω = B(0, 1); ϕ = ϕ(r)→ ψ(r) = r (N−2)/2ϕ(r).

||ϕ||H =[ ∫ 1

0|ψ′(r)|2r dr

]1/2.

Over the space of radially symmetric functions

−∆− λ∗|x |2

I in R3 ∼ −∆ in R2.

This guarantees coercivity in Hs , for 0 < s < 1.

When λ > λ∗ this transformation yields

−ψ′′ − ψ′

r− c

ψ

r 2= f ,

with c > 0. Consequently we have a non-admissible perturbationof the 2-d Laplacian. The equation does not make sense in thecontext of distributions....



The Dirichlet problem for the parabolic operator: Three cases.

ut −∆u − λ

|x |2 u = 0 in Ω× (0,∞)

u = 0 on Γ× (0,∞)u(x , 0) = u0(x) ∈ L2(Ω) in Ω.

0 < λ < λ∗: u∈ C([0,∞)] ; L2(Ω)

)∩ L2

(0,∞; H1

0 (Ω)).

λ = λ∗: u∈ C([0,∞)] ; L2(Ω)

)∩ L2(Ω) (0,∞;H(Ω)) .

λ > λ∗: Lack of well-posedness.

Furthermore, in the first two cases the L2-norm of solutionsdecays exponentially.Solutions have to be interpreted in the semigroup sense.Uniqueness does not hold in the distributional one. For instance,for

λ = λ∗, u(x) = |x |−(N−2)/2log(1/|x |),

is a singular stationary solution. It is not the semigroup solution.Enrique Zuazua Hardy inequalities and asymptotic behavior for heat kernels


The Cauchy problem

Consider now the Cauchy problem in the whole spaceut −∆u − λ

|x |2 u = 0 in RN × (0,∞)

u(x , 0) = u0(x) ∈ L2(RN) in Ω.

When λ ≤ λ∗ the problem is well-posed because of the Hardyinequality. The equation generates a semigroup of contractions inL2(RN).But, is there any decay rate? The classical Hardy inequalitydoes not answer to this question because of the lack ofHardy-Poincare version.To overcome this difficulty we perform the similaritytransformation:

w(y , s) = tN/4u(t1/2y , t); s = log(t + 1).

The equation then reads:

ws −∆y w − 1

2y · ∇w − N

4w − λ w

|y |2= 0.



The heat equation in the similarity variables is well-behaved in theweighted spaces L2(K ) =

∫f 2(y)K (y)dy <∞ (see M.

Escobedo & O. Kavian, 1987, ...), with K (y) = exp(|y |2/4).Recall that the classical Gaussian heat kernel is

G (x , t) = (4πt)−d/2 exp(−|x |2/4t)

so that the L2-norm decays at as t−d/4.The similarity scaling is the one dictated by the heat kernel.In the similarity variables, the gaussian heat kernel becomes astationary solution:

S(y) = (4π)−d/2 exp(−|y |2/4).



We prove the following sharp Hardy-Poincare inequality in theseweighted spaces:

N + 2

4

∫f 2Kdy +

(N − 2)2

4

∫f 2

|y |2Kdy ≤

∫|∇f |2Kdy

This yields the exponential decay rate for the evolution in similarityvariables in L2(K ) even for λ = λ∗. Returning to the originalvariables, for N ≥ 3 and λ = λ∗ we get:

||u(t)||L2(RN) ≤ Ct−1/4||u0||L2(K).

Thus, the above decay rate is the same as in dimension d = 1.The critical Laplacian-Hardy operator still diffuses as the laplacianin 1− d ...



Boundary singularities

Similar problems arise when the singularity is on the boundary:x = 0 ∈ ∂Ω. The same results apply. But there is there room forimprovement of the Hardy inequalities in that case?Consider first the half space Ω = RN

+ . In spherical harmonics thiscorresponds to considering functions that oscillate in the angularvariables, not radially symmetric, and therefore involving onlyhigher modes in the spherical harmonics decomposition. Actually,it is well known (Tertikas-Filippas-Tidblom, 2009) that, for all N:

N2

4

∫RN

+

|u|2

|x |2dx ≤

∫RN

+

|∇u|2dx .

This shows that the Hardy constant ”jumps” from (N − 2)2/4 toN2/4, when the singularity of the potential reaches the boundary.The same result holds smooth convex domains.



The overall picture depends on the local+ global properties of thedomain: 3

3C.Cazacu and E. Zuazua, “ Hardy inequalities and controllability of thewave equation with boundary quadratic singular potential”.



Twisted domains

Straight cylinder versus twisted one:

The cylinder Ω = ω × R and the twisted domain Ωθ, in which thecross section ω is twisted with angle θ depending on the parameterof the axis x3.In the cylinder:

−∆ΩθD − λ1 ≥ 0

λ1 being the first eigenvalue of the Dirichlet Laplacian in the crosssection ω.



Twisting gives Hardy inequalities and thus further decay rates.

In the twisted case the following Hardy inequality holds 4

−∆ΩθD − λ1 ≥ c

1

1 + x23

.

In a joint paper with D. Krejcirik 5 we show, using a carefulcombination of the analytical effects of twisting and similaritytransformations, that the heat semigroup gains a decay rate of theorder of t−1/2 in L2, because of twisting.

4T. Ekholm, H. Kovarik and D. Krejcirik, A Hardy inequality in twistedwaveguides, Arch. Ration. Mech. Anal. 188 (2008), 245–264.

5D. Krejcirik & E. Z., The Hardy inequality and the heat equation intwisted tubes, JMPA, to appear.



Consider the heat equation in the twisted domain

ut −∆u − λ1u = 0 in Ωθ × (0,∞).

Then, because of the positivity of the operator −∆− λ1I , we have

||u(t)||L2(Ωθ) ≤ ||u0||L2(Ωθ), ∀t ≥ 0.

But:

Does twisting affect the behavior as t →∞?

Does the Hardy inequality produce a decay rate?

Our main result confirms that holds:



We define the polynomial decay rate as follows:Γ(Ωθ,K ) :=

sup

Γ∣∣∣ ∃CΓ > 0, ∀t ≥ 0, ||SΩθ

D (t)||L2(Ωθ,K)→L2(Ωθ) ≤ CΓ (1 + t)−Γ

with K (x) := ex21/4

Theorem

(D. Krejcirik & E. Z., JMPA, to appear)

Γ(Ωθ,K )

= 1/4 if Ωθ is untwisted

≥ 3/4 if Ωθ is twisted.

3

4=

1

4+

1

2



Sketch of the proof:

1. Straightening of the tube Lθ : Ω0 → Ωθ (curvilinear

coordinates)

ut − (∂1 − θ ∂τ )2u −∆′u − λ1u = 0, in Ω0 × (0,∞)

Lθ(x) =

x1

x2 cos θ(x1) + x3 sin θ(x1)−x2 sin θ(x1) + x3 cos θ(x1),

∂τ := x3∂2 − x2∂3, ∆′ := ∂2

2 + ∂23



2. Similarity transformation.

u(y1, y2, y3, s) = es/4 u(es/2y1, y2, y3, e

s − 1︸︷︷︸t

)

us −1

2y1∂1u − (∂1 − σs ∂τ )2u − es∆′u − λ1es u − 1

4u = 0

σs(y1) := es/2θ(es/2y1)

||u(t)||L2(Ω0) = ||u(s)||L2(Ω0)



3. Changing the space. L2(Ω0,K ), K (y) = ey21 /4 =⇒

compactness of embeddings and discrete Fourier seriesrepresentation.4. Asymptotic analysis.

As s →∞, |σs(y1)| −→∣∣∣θ∣∣∣

L1(R)δ(y1); R× e−s/2ω −→ R

Projecting the equation

us −1

2y1∂1u − (∂1 − σs ∂τ )2u − es∆′u − λ1es u − 1

4u = 0

into the first eigenfunction with respect to the cross-section we get

ws −1

2y1∂1w − (∂1 − σs ∂τ )2w − 1

4w = 0

which singularly converges to

ϕs − 12 y1ϕy1 − ϕy1y1 − 1

4 ϕ = 0; Dirichlet b.c. at y1 = 0

iff Ωθ is twisted.Enrique Zuazua Hardy inequalities and asymptotic behavior for heat kernels


The tube has been broken into two semi-infinite tubes with aDirichlet boundary condition

What is the effect of a Dirichlet boundary condition on thehalf-line?We simply pass from the 1− d gaussian heat kernel

G (x , t) = (4πt)−1/2 exp(−|x |2/4t)

to its x-derivative

∂xG (x , t) = − 1√t

(4πt)−1/2 x

2√

texp(−|x |2/4t)

The L2 decay rate goes from 1/4 to 1/4 + 1/2.

Theorem

3

4=

1

4+

1

2



−5 0 5

−1

0

1

−5 0 5

−1

0

1



Similar phenomena arise in other geometric contexts:

Straight strip versus twisted one:

Twisting Dirichlet and Neumann boundary conditions on a strip:



What about twisted graphs?



Open problems:

Analyze in detail the linearization process. Back tononlinear....

Hardy inequalities, graphs, twisting and decay of heat kernels.

Elliptic operators involving singular first order terms.

Multipolar singularities (existing works by J. Dolbeault, M. J.Esteban,...).

Wave equation: Better explain the propagation phenomenausing bicharacteristic rays (semi-classical, Wigner,H-measures,...) and more geometrical tools.

Further analyze the effect of twisting and other geometricdeformations such as bending. Links with the theory of rods,shells,...?



Thank you!


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Hardy inequalities and asymptotic behavior for heat kernels