conformal geometry and elliptic operatorssystems/goldsteinv.pdfdomains under some additional...

Main problemsConformal mappings

Lower estimates for Neumann eigenvaluesExamples

Estimates for p-Laplace operatorSpectral stabilitySpace domains

Appendix

Conformal Geometry and Elliptic Operators

Vladimir Gol’dshteinBen-Gurion University of the Negev

Sde Boker, 2017March 20-24

0(Joint works with Viktor Burenkov and Alexander Ukhlov)Vladimir Gol’dshtein Ben-Gurion University of the Negev Conformal Geometry and Elliptic Operators




Appendix

This talk devoted to connections between conformal (hyperbolic)geometry of bounded simply connected planar domains and theNeumann eigenvalues of the p-Laplace operator (1 < p ≤ 2). Themain results concerned to a large class of bounded non convexdomains under some additional restrictions on its conformal(hyperbolic) geometry. This new class of planar domains includesquasidiscs (images of the unit disc under quasiconformalhomeomorphism of the plane). Situation in space domains will bediscussed also.

Vladimir Gol’dshtein Ben-Gurion University of the Negev Conformal Geometry and Elliptic Operators




Appendix

The two main problems will be discussed:

1. Lower estimates for the principal Neumann eigenvalues of the(p)-Laplace operator.

2. Spectral stability for Neumann-Laplace operators.





Appendix

The classical Neumann-Laplace spectral problem is:

−∆u = µu in Ω,

∂u∂n

∣∣∣∣∂Ω

= 0.

The weak statement of the spectral problem is: a function

u ∈W 1,2(Ω) solves the previous problem iff

∫

Ω

∇u(x) · ∇v(x) dvol = µ

∫

Ω

u(x)v(x) dvol , ∀v ∈ W 1,2(Ω).

Here W 1,2 is the space of integrable functions with the bounded

energy integral∫Ω |∇u(x)|2dvol.





Appendix

By the Max-Min principle the first nontrivial Neumann eigenvalueof the Laplace operator µ1 can be characterized as

µ1(Ω) = min

∫Ω

|∇u(x)|2 dvol

∫Ω

|u(x)|2 dx: u ∈W 1,2(Ω) \ 0,

∫

Ω

u dvol = 0

.





Appendix

It means that the µ1(Ω)−

12 is the best constant B2,2(Ω) in the

following Sobolev-Poincare inequality

infc∈R‖f − c | L2(Ω)‖ ≤ B2,2(Ω)‖∇f | L2(Ω)‖, f ∈W 1,2(Ω).





Appendix

Recall that the Sobolev space W 1,p(Ω), 1 ≤ p <∞, is defined asa Banach space of locally integrable weakly differentiable functionsf : Ω→ R equipped with the following norm:

‖f |W 1p (Ω)‖ =

(∫

Ω

|f (x)|p dvol) 1

p

+

(∫

Ω

|∇f (x)|p dvol) 1

p

.

The previous assertions are correct for the first nontrivial Neumanneigenvalue of the (p)-Laplace operator in terms of W 1,p-spaces.





Appendix

Short historic remarks for the Neuman-Laplace operator

1. Lord Rayleigh, The theory of sound, London, 1894/96(formulation of the spectral problem)

2. H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwertelinearer partieller Differentialgleichungen, Math. Ann. 71,1912. (Asymptotic of eigenvalues).

3. G. Polya, G. Szego, Isoperimetic inequalities in mathematicalphysics, Princeton University Press, 1951. (Upper estimates)

4. L. E. Payne, H. F. Weinberger, An optimal Poincar inequalityfor convex domains, Arch. Rat. Mech. Anal., 5, 1960. (Lowerestimates for convex domains in terms of its diameters)





Appendix

The classical results by L. E. Payne and H. F. Weinberger (1960)give the lower estimates of the first non-trivial eigenvalue of theNeumann Laplacian in convex domains in terms of its diameters:

µ1[Ω] ≥π2

d(Ω)2.





Appendix

Infinitesimally a conformal homeomorphism is a similarity. It meansthat

∇(f (ϕ(x , y)) = (∇f )(ϕ(x , y))|ϕ′

z(x , y)

for any conformal homeomorphism ϕ : Ω→ Ω′ and any smoothfunction f : Ω′ → R . Hence |ϕ′

z(x , y)|2 = J((x , y);ϕ). Thisequality can be used as an alternative definition of conformalmappings. Here z = x + iy .





Appendix

Basic Lemma. Any conformal mapping ϕ : Ω→ Ω′ preserves the

energy integral:

∫

Ω|∇(f ϕ)|2dvol =

∫

Ω|∇(f ϕ)|2 |ϕ

′

z(x , y)|2|J((x , y);ϕ)| |J((x , y);ϕ)|dvol

=

∫

Ω′

|∇f |2dvol





Appendix

The quantity

Q := sup(x ,y)∈Ω|Dϕ(x , y)|2|J((x , y);ϕ)|

is called a coefficient of quasiconformality (a quasiconformaldilatation).If Q is bounded a corresponding homeomorphism is calledquasiconformal. It is one of the classical definitions (byB.V.Shabat). Quasiconformal homeomorphisms quasi-preserve theenergy integral.





Appendix

Let us give simple illustration to our method. Consider the ellipse

E ⊂ R2: (x , y) ∈ R2 : x2

a2+ y2

b2≤ 1, a ≥ b. The linear mapping

ϕ(x , y) =

(a 00 b

)(a

b

)

maps the unit disc D onto E with the qusiconformality coefficientQ(D) = b

a. Using this change of variable in Sobolev-Poincare

inequality we obtain µ1(E ) ≥(j ′1,1)

2

a2. This estimate is new and it is

better then the classical estimate for convex domainsµ1(E ) ≥ π2

d(E)2, because d(E ) = 2a and 2j ′1,1 > π, j ′1,1 ≈ 1.84118.





Appendix

The suggested method is based on the following diagram proposedin (V.G. and L. Gurov, 1994, V.G. and A. Ukhlov, 2009).

W 1,p(Ω)(ϕ−1)∗−→ W 1,q(D)

↓ ↓Ls(Ω)

ϕ∗

←− Lr (D)

Here the operator ϕ∗ is a bounded composition operator onLebesgue spaces induced by a homeomorphism ϕ of Ω and D andthe operator (ϕ−1)∗ is a bounded composition operator on Sobolevspaces.





Appendix

We suggest the estimates in terms of the hyperbolic radii for alarge class of bounded non-convex domains with some additionalrestrictions on the hyperbolic geometry that we call a conformalregularity:

A simply connected planar domain Ω with non-empty boundary iscalled a conformal α-regular domain if there exists a Riemannmapping ϕ : Ω→ D:

∫

D

|(ϕ−1)′(w)|α dvol <∞ for some α > 2.





Appendix

In the case of conformal α-regular domains Ω ⊂ C we haveembedding

Lr (Ω, h) → Ls(Ω), s =α− 2

αr :





Appendix

Theorem A. Let Ω ⊂ R2 be a conformal α-regular domain. Then

the spectrum of the Neumann-Laplace operator in Ω is discrete,

can be written in the form of a non-decreasing sequence

0 = µ0[Ω] < µ1(Ω) ≤ µ2(Ω) ≤ ... ≤ µn(Ω) ≤ ... , and

1/µ1(Ω) ≤4

α√π2

(2α− 2

α− 2

) 2α−2α

‖ψ′ | Lα(D)‖2

≤ 4α√π2

(2α− 2

α− 2

) 2α−2α

∫

D

R2Ω(ψ(x))

(1− |x |2)2 dvol

2α

(1)

where ψ : D→ Ω is the Riemann conformal mapping of the unit

disc D ⊂ R2 onto Ω and RΩ(ψ(x)) is a hyperbolic radius of Ω.





Appendix

A K -quasidisc is the image of the unit disc under aK -quasiconformal mapping of the plane onto itself.Theorem for Quasidiscs. Suppose a conformal homeomorphism

ϕ : D→ Ω maps the unit disc D onto a K-quasidisc Ω. Then

1/µ1[Ω] ≤ B22α/(α−2),2[D]

∫

D

|ϕ′(x , y)|α dvol

2α

≤ 4π−2α

(2α− 2

α− 2

) 2α−2α

‖ϕ′ | Lα(D)‖2

for any 2 < α < 2K2

K2−1.





Appendix

Consider the interior of the cardioid (it is not a quasidsic).Example. Let Ωc be the interior of the cardioid. The

diffeomorphism

z = ψ(w) = (w + 1)2, z = x + iy ,

is conformal and maps the unit disc D onto Ωc . Then by Theorem

A

‖ψ′ | L∞(D)‖ = maxw∈D

2|w + 1| ≤ 4.

Hence

µ1(Ωc) ≥(j

′

1,1)2

16.

Here j1,1 is the first positive zero of the derivative of the Bessel

function J′

1.





Appendix

The quantity

Qp = sup(x)∈Ω|Dϕ(x)|p|J(x);ϕ)|

is called a p-dilatation and corresponding homeomorphisms arecalled p-quasiconformal (or homeomorphisms with boundedp-dilatation). These classes is comparatively new and wereintroduced in 1994 by G, L.Gurov and A.Romanov in theframework of the composition operator theory for Sobolev spacesW 1

p .





Appendix

The variational formulation of spectral problems for the p-Laplaceoperator is based on the Dirichlet (energy) integrals

‖u | L1p(Ω)‖p :=

∫

Ω

|∇u(x)|p dvol .

The p-quasiconformal homeomorphisms induce boundedcomposition operators for such energy integrals and we use themand their generalizations for estimates of spectrum in roughdomains.





Appendix

Brennan’s conjecture

is that for a conformal mapping ϕ : Ω→ D

∫

Ω

|ϕ′(x , y)|δ dvol < +∞, for all4

3< δ < 4. (2)

For the inverse conformal mapping ψ = ϕ−1 : D→ Ω Brennan’sconjecture states

∫∫

D

|ψ′(u, v)|γ dvol < +∞, for all − 2 < γ <2

3. (3)

For bounded domains −2 < γ ≤ 2. The upper bound γ = 2 isaccurate (V.G. and A. Ukhlov, 2009).





Appendix

Theorem B. Let ϕ : D→ Ω be a conformal homeomorphism from

the unit disc D to a conformal α-regular domain Ω and Brennan’s

Conjecture holds. Then for every p ∈ (max4/3, (α+ 2)/α, 2)the following inequality is correct

1

µp(Ω)≤ 2

32 (2π)

(α−2)q−αp

αq ·‖(ϕ)′|Lα(D)‖2∫

D

|(ϕ−1

)′ |(p−2)qp−q dvol

p−q

q

for any q ∈ [1, 2p/(4− p)).





Appendix

Here Br ,q(D) ≤ 232 (2π)

1r−

1q is the best constant in the

Sobolev-Poincare inequality in the unit disc D ⊂ C and Kp,q(Ω) isthe norm of composition operator

(ϕ−1

)∗: L1,p(Ω)→ L1,q(D)

generated by the inverse conformal mapping ϕ−1 : D→ Ω:

Kp,q(Ω) ≤

∫

D

|(ϕ−1

)′ |(p−2)qp−q dvol

p−q

pq

.





Appendix

DefinitionConformal regular domains Ω1,Ω2 are conformal regular equivalentdomains if there exists a conformal mapping ψ : Ω1 → Ω2 suchthat∫∫

Ω1

|(ψ′(z)|α dxdy <∞ &

∫∫

Ω2

|(ψ−1)′(w)|α dudv <∞ (4)

for some α > 2.





Appendix

In the conformal regular planar domains Ω ⊂ C the spectrum ofthe Neumann Laplacian is discrete and can be written in the formof a non-decreasing sequence

0 < µ1(Ω) ≤ µ2(Ω) ≤ ... ≤ µn(Ω) ≤ ... ,

where each eigenvalue is repeated as many times as its multiplicity.





Appendix

Stability Theorem Let Ω1,Ω2 ⊂ C be conformal regularequivalent domains. Then for any n ∈ N

|µn(Ω1)−µn(Ω2)| ≤ 2cn

[C

(4α

α− 2

)]2Eα(ϕ1, ϕ2)‖ϕ1−ϕ2 | L1,2(D)‖ ,

(5)where Ω1 = ϕ1(D), Ω2 = ϕ2(D) and

cn = maxµ2n(Ω1), µ2n(Ω2) . (6)





Appendix

Here

Eα(ϕ1, ϕ2) =

∫

D

max

|ϕ′

1(z)|α|ϕ′

2(z)|α−2,|ϕ′

2(z)|α|ϕ′

1(z)|α−2

dvol

1α

<∞





Appendix

Theorem C. Suppose that there exists a 2-quasiconformalhomeomorphism ϕ : Ω→ Ω, of a bounded Lipschitz domainΩ ⊂ R

n onto Ω, such that

M2(Ω) = ess supx∈Ω|J(x , ϕ)|

12 <∞.

Then the spectrum of Neumann-Laplace operator in Ω is discrete,can be written in the form of a non-decreasing sequence

0 = µ0(Ω) < µ1(Ω) ≤ µ2(Ω) ≤ ... ≤ µn(Ω) ≤ ... ,

and1

µ1(Ω)≤ K 2

2 (Ω)M22 (Ω)

1

µ1(Ω).





Appendix

Denote by H1 the standard n-dimensional simplex, n ≥ 3,

H1 := x ∈ Rn : n ≥ 3, 0 < xn < 1, 0 < xi < xn, i = 1, 2, . . . , n−1.

Theorem D. Let

Hg := x ∈ Rn : n ≥ 3, 0 < xn < 1, 0 < xi < xγin , i = 1, 2, . . . , n−1

γi ≥ 1, γ := 1 +∑n−1

i=1 γi , g := (γ1, ..., γn−1) .Then the spectrum of the Neumann-Laplace operator in the

domain Hg is discrete, can be written in the form of a

non-decreasing sequence

0 = µ0(Hg ) < µ1(Hg ) ≤ µ2(Hg ) ≤ ... ≤ µn(Hg ) ≤ ... ,

and for any r > 2 the following inequality holds:





Appendix

1

µ1(Hg )≤

infa

(a2(γ21 + ...+ γ2n−1 + 1)− 2a

n−1∑

i=1

γi

)

a

(∫

H1

(xaγ−nn

) rr−2 dvol

) r−2r

B2r ,2(H1),

where (2n)/(γr) < a ≤ (n − 2)/(γ − 2) and Br ,2(H1) is the bestconstant in the (r , 2)-Sobolev-Poincare inequality in the domainH1.





Appendix

Recall the analytic description of homeomorphisms which generatebounded composition operators (A. Ukhlov, 1993):Composition Theorem A homeomorphism ϕ : Ω→ Ω′ between

two domains Ω,Ω′ ⊂ Rn, n ≥ 2, induces a bounded composition

operator

ϕ∗ : L1p(Ω′)→ L1q(Ω), 1 ≤ q < p <∞,

if and only if ϕ ∈W 11,loc(Ω), has finite distortion, and

Kp,q(f ; Ω) =

(∫

Ω

( |Dϕ(x)|p|J(x , ϕ)|

) q

p−q

dx

) p−q

pq

<∞.





Appendix

If p = q then the sufficient and necessary analytic condition is:

Kp(f ; Ω) = ess supx∈Ω

|Dϕ(x)|p|J(x , ϕ)| <∞.

( Vodop’yanov and G., 1975, G., Gurov, Romanov 1995)In the case p = n we have the definition of mappings of boundeddistortion and we call mappings that generate boundedcomposition operators as mappings of bounded (p, q)-distortion.

The homeomorphisms that generate bounded compositionoperators of Sobolev spaces L11(Ω

′) and L11(Ω) were introduced byV. G. Maz’ya (1969) as a class of sub-areal mappings. Thispioneering work established a connection between geometricalproperties of homeomorphisms and corresponding Sobolev spaces.





Appendix

V. I. Burenkov, V. Gol’dshtein, A. Ukhlov, Conformal spectralstability for the Dirichlet-Laplace operator, Math. Nachr., 288(2015), 1822–1833.

V. I. Burenkov, V. Gol’dshtein, A. Ukhlov, Conformal spectralstability estimates for the Neumann Laplacian. Volume 289,Issue 17-18, December 2016 Pages 21332146.

V. Gol’dshtein, A. Ukhlov, On the First Eigenvalues of FreeVibrating Membranes in Conformal Regular Domains, RationalMech Anal (2016) 221: 893, DOI:10.1007/s00205-016-0988-9.

V. Gol’dshtein, A. Ukhlov, Spectral estimates of the p-LaplaceNeumann operator in conformal regular domains. Transactionsof A. Razmadze Mathematical Institute Volume 170, Issue 1,May 2016, Pages 137148.Vladimir Gol’dshtein Ben-Gurion University of the Negev Conformal Geometry and Elliptic Operators




Appendix

THANKS


conformal geometry and elliptic operatorssystems/goldsteinv.pdfdomains under some additional...

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