[PDF] - Composition operators on Sobolev spaces* A. Ukhlov Ben-Gurion PDF Document

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Composition operators on Sobolev spaces*

Ben-Gurion University of the Negev, Israel (*joint works with V. Gol’dshtein and S. K. Vodop’yanov) The Dirichlet problem and composition operators We study composition operators on Sobolev spaces of weakly differentiable functions. Namely, we study operators ϕ∗ : L1

1 ≤ q ≤ p ≤ ∞, which are defined by the composition rule: ϕ∗(f) = f ◦ ϕ, f ∈ L1

Here ϕ is a mapping ϕ : Ω → Ω′ of the Euclidean domains Ω, Ω′ ⊂ Rn, n ≥ 2. The Sobolev space L1

equipped with the following seminorm: f | L1

|∇f|p(x) dx 1/p , 1 ≤ p < ∞, f | L1

|∇f| where ∇f is the weak gradient of the function f.

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The composition operators theory is closely connected with the elliptic equations theory. For example, consider the classical Dirichlet problem for the Laplace operator in the plane domain Ω ⊂ R2. ∆u = 0, u|∂Ω = f. u ∈ C2(Ω), ∆u = ∂2u

It is well known that for the unit disc D ⊂ R2 we have the unique solution of this problem u(r0, θ0) = 1 2π

1 + r2

(the Poisson formula), obtained by means composition of the av- erage value function with the conformal mapping ϕ : D → D.

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If Ω be a connected simply connected plane domain with non- empty boundary then by the Riemannian Mappings Theorem there exists a conformal homeomorphism ϕ : Ω → D. Recall that a mapping ϕ = u(x, y) + iv(x, y) : Ω → D is called a conformal homeomorphism if ϕ is differentiable (ana- lytic) in Ω and ϕ′(z) = 0, z ∈ Ω. It is well known that a composition of an analytic function and harmonic function is an harmonic function again. So, if a function u is a solution of the Dirichlet problem for the unit disc, then a function u ◦ ϕ is a solution of the Dirichlet problem in the domain Ω ⊂ R2 (with a corresponding boundary value). Let Ω = {(x, y) : y > 0} (upper half of the plane R2) and u(x, 0) = f(x). The conformal mapping w = ϕ(z) = z−z0

solution is given by u(x, y) = 1 π

y (t − x)2 + y2f(x)dx.

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In analytical terms the condition of conformity we can rewrite as: ϕ : Ω → D, is a smooth mapping such that |Dϕ(x)|2 = |J(x, ϕ)|, x ∈ Ω. Here Dϕ is the Jacobi matrix of ϕ and J(x, ϕ) = det(Dϕ(x)). Note, that conformal mappings preserve the Dirichlet integral:

|∇u ◦ ϕ)|2 dz =

|∇u|2 dw and we can consider solution of the Dirichlet problem as an applica- tion of the Dirichlet’s principle to the energy integral (R. Courant. ”Dirichlet principle, Conformal mappings and Minimal Surfaces”). In another words we can say that: Conformal mappings generate a bounded composition operator (isomorphism) ϕ∗ : L1

ϕ∗(u) = u ◦ ϕ. Here L1

functions with finite Dirichlet integral.

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The boundary value problem for the p-Laplace operator ∆pu = div(|∇u|p−2∇u) leads to composition operators in more general Sobolev spaces, namely, L1

many authors (V. Gol’dshtein, S. Hencl, P. Koskela, I. Markina,

The theory of composition operators on Sobolev spaces arises to the Yu. G. Reshetnyak problem (1968) about description of all isomorphisms ϕ∗ of homogeneous Sobolev spaces L1

quasiconformal mappings ϕ of Euclidean spaces Rn, n ≥ 2, by the composition rule ϕ(f) = f ◦ ϕ.

homeomorphism ϕ : D → D′ induces a composition operator ϕ∗ from L1

is also quasiconformal one then ϕ∗ is an isomorphism of L1

and L1

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Recall that a homeomorphism ϕ : Ω → Ω′ is called quasiconfor- mal if ϕ belongs to the Sobolev class W 1,1

constant 1 ≤ K < ∞ such that |Dϕ(x)|n ≤ K|J(x, ϕ)| for almost all x ∈ Ω. For p = n a homeomorphisms ϕ : D → D′ induces an iso- morphisms of Sobolev spaces W 1

if ϕ it is a bi-Lipschitz one. This result was proved for p > n

by V. Gol’dshtein and A. S. Romanov (1984) and for 1 ≤ p < n by I. Markina (1990). Homeomorphisms that induce bounded composition operators from Sobolev space L1

nov (1988). A geometric description of such homeomorphisms was

manov (1995). The multipliers theory was applied to the composition operators theory by V. G. Maz’ya and T. O. Shaposhnikova (1986).

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New effects arise when we study composition operators with de- creasing summability. In 1993 by A. Ukhlov was proved: A homeomorphism ϕ : Ω → Ω′ generates by the composition rule ϕ∗f = f ◦ ϕ the bounded operator ϕ∗ : L1

if and only if ϕ ∈ L1

Kp,q(f) =

|Dϕ|p |J(x, ϕ)| q

dx p−q

< ∞. A mapping ϕ : Ω → Rn belongs to L1

functions ϕj belong to L1

Jacobi matrix Dϕ(x) = ∂ϕi

nant (Jacobian) J(x, ϕ) = det Dϕ(x) are well defined at almost all points x ∈ Ω. The norm |Dϕ(x)| of the matrix Dϕ(x) is the norm

the matrix Dϕ(x). We will use the same notation for this matrix and the corresponding linear operator. A mappings ϕ : Ω → Rn has a finite distortion if Dϕ(x) = 0 for almost all points x that belongs to set Z = {x ∈ D : J(x, ϕ) = 0}.

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Necessity of studying of Sobolev mappings with integrable distor- tion arises in problems of the non-linear elasticity theory. J. M. Ball (1976, 1981) introduced classes of mappings, defined on bounded domains Ω ∈ Rn: A+

{ϕ ∈ W 1

p, q > n, where adj Dϕ is the formal adjoint matrix to the Jacobi matrix Dϕ: adj Dϕ(x) · Dϕ(x) = I ·J(x, ϕ). Mappings that generate composition operators on Sobolev spaces are mappings of finite distortion. The theory of mappings of fi- nite distortion is under intensive development at the last decades. In series of works the geometrical and topological properties of these mappings were studied (S. Hencl, J. Heinonen, I. Holopainen,

y, J. Manfredi, G. Martin, O. Mar- tio, P. Pankka, V. Ryazanov, U. Srebro, V. ˇ Sver´ ak, E. Villamor,

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Mappings that generate bounded composition operators on So- bolev spaces ϕ∗ : L1

are natural generalization of (quasi)conformal mappings (mappings

(p, q)-distortion. In the description of composition operators on Sobolev spaces the significant role belongs the additive set function which allows to localize the composition operators. Let a mapping ϕ : Ω → Ω′, where Ω, Ω′ are domains of Eu- clidean space Rn, generates a bounded composition operator ϕ∗ : L1

1 ≤ q < p ≤ ∞. Then Φ(A′) = sup

, where number r is defined according to 1/r = 1/q − 1/p, is a bounded monotone countably additive function defined on open bounded subsets A′ ⊂ Ω′.

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The description of the composition operators is based on the following local estimate: |Dϕ(x)|q ≤

p−q

where Φ′ is the volume derivative of the set function Φ. Let a monotone finitely additive set function Φ be defined on

x ∈ Ω the volume derivative Φ′(x) = lim

Φ(Bδ) |Bδ| is finite and for any open set U ⊂ Ω, the inequality

Φ′(x) dx ≤ Φ(U) is valid.

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It is well known that the mapping inverse to a quasiconformal homeomorphism is quasiconformal also. We have the similar result for homeomorphisms with bounded (p, q)-distortion. Let ϕ : Ω → Ω′ be a homeomorphism with bounded (p, q)- distortion, q > n − 1. Then the inverse mapping ϕ−1 : Ω′ → Ω is a mapping with bounded (q′, p′)-distortion, q′ = q/(q − n + 1) and p′ = p/(p − n + 1). The main difficult here is to prove that the mapping which is inverse to a homeomorphism with bounded (p, q)-distortion is the Sobolev mapping.

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The very interesting case in the composition operators theory arises when p = ∞. In this case we have A homeomorphism ϕ : Ω → Ω′ between two domains Ω, Ω′ ⊂ Rn belongs to the Sobolev space L1

the composition operator ϕ∗ : L1

is bounded. The proof of this fact is based on Let a function f : Ω → R belong to the Sobolev space L1

Then there exists a sequence of smooth functions {fk} ∈ L1

such that the sequence ∇fk weakly converges to ∇f in L∞(Ω).

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The next theorem asserts regularity of the inverse mapping in this case: Let a homeomorphism ϕ : D → D′ between two domains D and D′ ⊂ Rn, n ≥ 2, generate by the composition rule ϕ∗f = f ◦ ϕ a bounded operator ϕ∗ : L1

possess the Luzin N-property and have finite distortion. Then the inverse mapping ϕ−1 : D′ → D generates the composition

(ϕ−1)∗ : L1

and belongs to the Sobolev space L1

The theorem is closely connected with the problem of regularity

sively studied at the last time (M. Cs¨

The proof of the theorem is based on the lower estimate of the composition operator f | L1

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As an application of composition operators theory we study the Sobolev type embedding theorems in weighted Sobolev spaces. This part of my talk is based on recent article, joint with V. Gol’dshtein and V. V. Motreanu. Let Ω be a domain in the Euclidean space Rn. For p ∈ (1, +∞), q ∈ [1, +∞) we define the two-weighted Sobolev space W 1,q,p(Ω, v, w) = {u ∈ Lq(Ω, v) : ∇u ∈ Lp(Ω, w)} equipped with the norm uW 1,q,p(Ω,v,w) = uLq(Ω,v) + |∇u| Lp(Ω,w), where ∇u is the weak gradient of the function u, i. e. ∇u = ( ∂u ∂x1 , ..., ∂u ∂xn ).

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We have the following motivation for these spaces: if a function u belongs to the usual Sobolev space W 1,p(Ω), 1 ≤ p < n, and Ω ⊂ Rn is a regular domain, then by the classical Sobolev embedding theorem we have that u ∈ Lq(Ω), q = np/(n−p). It means that in the case of regular domains the space W 1,p(Ω), 1 ≤ p < n, really is the space W 1,q,p(Ω). So, the main question here is the existence

We will use also the seminormed space L1

equipped with the seminorm uL1

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We suppose that weight functions v : Rn → R and w : Rn → R satisfying the Muckenhoupt Aq-condition and Ap-condition respec-

condition if sup

1 |B|

w dx 1 |B|

w

q−1 < +∞. For such weights the Lebesgue space Lq(Ω, v) and Sobolev space W 1,q,p(Ω, v, w) are Banach spaces. Let C∞

in Ω. Note that it is a subspace of W 1,q,p(Ω, v, w). Then, we define W 1,q,p (Ω, v, w) as the closure of C∞

In the particular case where w = w0 and q = p, we retrieve the usual weighted Sobolev spaces W 1,p(Ω, w) := W 1,p,p(Ω, w, w) and W 1,p

(Ω, w, w), studied for instance by T. Kilpelai- nen, A. Kufner and B. O. Turesson.

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Our approach to the weighted embedding theorems is based on the systematic application of the composition operators theory on weighted Sobolev spaces to the embedding theory. This approach was suggested by V. Gol’dshtein and V. Gurov for classical Sobolev spaces and can be briefly described with the help of the following diagram for weighted spaces: W 1,p(Ω, w)

− → W 1,p1(D) ↓ ↓ Lq(Ω, v)

← − Lq1(D) Here the operator ϕ∗f = f◦ϕ is a bounded composition operator

ϕ∗ : W 1,p(Ω) → W 1,p1(D) induced by a homeomorphism ϕ : D → Ω that maps smooth domain (an embedding domain) D ⊂ Rn onto non smooth do- main Ω ⊂ Rn. Suppose that its inverse homeomorphism ϕ−1 : Ω → D induces a bounded composition operator of corresponding Lebesgue spaces. If the Sobolev space W 1

(compact) embedding operator into Lq1(D) then, using the corre- sponding compositions, we can construct the embedding operator

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Composition operators on weighted spaces Let D and Ω be domains in Euclidean space Rn, n ≥ 2. Re- member that a homeomorphism ϕ : D → Ω belongs to Sobolev class W 1,1

note by Dϕ the weak differential of ϕ. The norm |Dϕ(x)| is the standard norm of the linear operator defined by Dϕ(x) and J(x, ϕ) = det(Dϕ). Let D and Ω be domains in Euclidean space Rn, n ≥ 2 and homeomorphism ϕ : D → Ω. The composition operator ϕ∗ : L1

is bounded if and only if ϕ ∈ W 1,1

|Dϕ(x)|v(x)

|J(x, ϕ)|w(ϕ(x)) q

dx p−q

= K < +∞. In the case v = 1 we call such mappings ϕ : D → Ω as w- weighted (p, q)-quasiconformal homeomorphisms.

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Embedding theorems On the base of composition operators on weighted Sobolev spaces we prove the embedding theorem for two-weighted Sobolev spaces. Let a domain D ⊂ Rn be an embedding domain and there exists a w-weighted (p, p1)-quasiconformal homeomorphism ϕ : D → Ω

If for some p ≤ q ≤ q1 < ∞ the following inequality is correct

q1

dx < +∞, then an embedding operator i : W 1,p(Ω, w) ֒ → Lq(Ω, v) is bounded, if q ≤ q1 ≤ np1/(n − p1), and is compact if q ≤ q1 < np1/(n − p1).

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Examples of domains with embedding properties We study a family of domains containing as particular cases domains with anisotropic H¨

Assume n ≥ 2. Let b > 0. Let gi : [0, +∞) → [0, +∞) (for i = 1, . . . , n − 1) be C1 functions with the following properties: (a) gi(0) = 0 and gi(t) > 0 for t > 0; (b) there are constants t0 ∈ (0, 1), γ, δ > 1 and M1, M2 > 0 such that M1tγ−1 ≤

gi(t) ≤ M2tδ−1 for all t ∈ (0, t0). A typical example for gi is gi(t) = tγi (for i = 1, . . . , n − 1) where γi ≥ 1. In this case, the above properties are satisfied with γ = δ = N−1

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Denoting g = (g1, . . . , gn−1), we consider the domain Hg,b = {x ∈ Rn : 0 < xn < b, 0 < xi < gi(xn), i = 1, . . . , n−1}. For g1(t) = . . . = gn−1(t) = t we will use the notation ˆ Hb instead of Hg,b. Every domain Hg,b (in particular, ˆ Hb) satisfies an embedding property since it is bounded. If the weight is polynomial, i.e. w(x) := |x|α, then w ∈ Ap(Rn) if −n < α < n(p − 1). Let p ∈ (1, +∞), q ∈ [1, +∞). Suppose that −n < α < n(p − 1), max{−n, −δ} < β < n(q − 1) (resp. max{−n, −δ} < β ≤ 0 for q = 1) and p < min{α + γ, n}. Then the embedding (L1,p

is compact whenever q < min

such q, the embedding (W 1,q,p (Hg,b, |x|β, |x|α), · L1,p(Hg,b,|x|α)) ⊂ Lq(Hg,b, |x|β) is compact. Note that, in the classical case (i.e., α = β = 0 and γ = δ = n) we retrieve the condition q < p∗ =

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The proof of the theorem is based on the mappings ϕa : ˆ Hb → Hg,b given by ϕa(x) = (x1 xn ga

xn ga

Dirichlet problems We study the existence of weak solutions for the following two problems: for p, q ∈ (1, +∞), (P1)

u ∈ W 1,q,p (Ω, v, w), and for p ∈ (1, +∞), q ∈ [1, +∞), (P2)

u ∈ W 1,q,p (Ω, 1, w), where f ∈ Lq′(Ω) (with q′ =

Such type problems arise in the non-linear elasticity theory.

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The operator ∆p,w entering problems (P1), (P2) is a w-weighted p-Laplacian: −∆p,w(u), f =

w(x)|∇u|p−2∇u · ∇f dx, ∀u, f ∈ W 1,q,p (Ω, v, w). We say that u ∈ W 1,q,p (Ω, v, w) is a weak solution of problem (P1) if

w(x)|∇u|p−2∇u · ∇f dx =

v(x)|u|q−2uf dx, ∀f ∈ W 1,q,p (Ω, v, w), and we say that u ∈ W 1,q,p (Ω, w) is a weak solution of problem (P2) if

w(x)|∇u|p−2∇u · ∇f dx =

fv dx, ∀f ∈ W 1,q,p (Ω, w).

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In order to show the existence of weak solutions for problems (P1), (P2), we will assume that the space W 1,q,p (Ω, v, w)) is com- pactly embedded in the Lebesgue space Lq(Ω, v) (with v = 1 in the case of problem (P2)). Under this assumption, we show that problem (P1) admits at least two nontrivial weak solutions of op- posite constant sign, and that problem (P2) admits a unique weak

for the usual Sobolev space W 1,p(Ω). We solve the Dirichlet boundary value problems (P1) and (P2) under the assumption that the domain Ω satisfies a suitable em- bedding property. More precisely, we show: Let p > 1, q ≥ 1 such that p = q. Assume w ∈ Ap(Rn) and v ∈ Aq(Rn). Suppose that we have a compact embedding (W 1,q,p (Ω, v, w), · L1,p(Ω,w)) ⊂ Lq(Ω, v). Then, the problem (P1)

u ∈ W 1,q,p (Ω, v, w), has at least two nontrivial weak solutions u1 ≥ 0, u2 ≤ 0.

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Let p > 1, q ≥ 1, f ∈ Lq′(Ω) and let w ∈ Ap(Rn). Suppose that we have a compact embedding (W 1,q,p (Ω, w), · L1,p(Ω,w)) ⊂ Lq(Ω). Then, the problem (P2)

u ∈ W 1,q,p (Ω, w), has a unique weak solution. The proof of these theorem is based on the theorem (M. Otani). Consider X a real Banach space and let X∗ be its topological dual space. By ·, · we denote the duality brackets for the pair (X, X∗). Let Φ(X) be the set of lower semicontinuous, convex functions φ : X → R∪{+∞} which are not identically +∞. The effective domain of φ ∈ Φ(X) is D(φ) = {v ∈ X : φ(v) < +∞}, and the subdifferential ∂φ of φ is ∂φ(v) = {v∗ ∈ X∗ : φ(w) − φ(v) ≥ v∗, w − v, ∀w ∈ D(φ)} with the domain D(∂φ) = {v ∈ X : ∂φ(v) = ∅}. Note that, if ϕ is of class C1 at v ∈ X, then ∂φ(v) = {ϕ′(v)}.

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Theorem. (M. Otani) Let X be a real Banach space and φ1, φ2 ∈ Φ(X) be two nonnegative functions satisfying: (i) there exist α1, α2 ≥ 1, α1 = α2 such that φi(λv) = λαiφi(v), ∀λ > 0, ∀v ∈ D(φi), i = 1, 2; (ii) there is a constant C > 0 such that φ2(v)1/α2 ≤ C φ1(v)1/α1, ∀v ∈ D(φ1). Suppose there is u ∈ D(∂φ2) such that (iii) u gives the best constant for (ii), i.e., R(u) = max{R(v) : v ∈ D(φ1), φ1(v) = 0}, where R(v) := φ2(v)1/α2

(iv) α1φ1(u) = α2φ2(u). Then u ∈ D(∂φ1) and ∂φ2(u) ⊂ ∂φ1(u). In particular, if ∂φ1 is single-valued, then u is a nontrivial solution of ∂φ1(u) = ∂φ2(u). For the solution of (P1) we will apply Theorem (M. Otani) with X = W 1,q,p (Ω, v, w) endowed with the norm · L1,p(Ω,w), and the functions φ1, φ2 : X → R given by φ1(v) = 1 p ∇vp

and φ2(v) = 1 q vq

Note that ϕ′

w(x)|∇u|p−2∇u·∇f dx, ∀u, f ∈ W 1,q,p (Ω, v, w). and ϕ′

v(x)|u|q−2uf dx, ∀u, f ∈ W 1,q,p (Ω, v, w).

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For the solution of (P2) we will apply Theorem (M. Otani) with X = W 1,q,p (Ω, w) endowed with the norm · L1,p(Ω,w) and the functions φ1, φ2 : X → R given by φ1(v) = 1 p ∇vLp(Ω,w) and φ2(v) =

fv dx

Functionals associated to problems (P1) and (P2) Let J1 : W 1,q,p (Ω, v, w) → R and J2 : W 1,q,p (Ω, w) → R be the functionals associated to problems (P1) and (P2) respectively, that is J1(u) = 1 p

w(x)|∇u|p dx − 1 q

w0(x)|u|q dx, ∀u ∈ W 1,q,p (Ω, v, w), J2(u) = 1 p

w(x)|∇u|p dx −