Applications of change of variables operators for exact embedding theorems

Abstract. In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A pcondition. Sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities. The proposed method is based on the concept of 'generalized' quasiconformal homeomorphisms (homeomorphisms with bounded mean distortion). The choice of the homeomorphism type depends on the choice of the corresponding weighted Sobolev space. Such classes of homeomorphisms induce bounded composition operators for weighted Sobolev spaces. With the help of these homeomorphism classes the embedding problem for nonsmooth domains is reduced to the corresponding classical embedding problem for smooth domains. Examples of domains with anisotropic Hölder singularities demonstrate the sharpness of our machinery comparatively with known results.

show abstract

“…Let us apply these general results to domains with anisotropic Hölder singularities described in [9].…”

Section: Proof Becausementioning

confidence: 98%

“…Here we adopt the scheme of [9] to the case of weighted Sobolev spaces. The next lemma is the main technical result of this section.…”

Section: Gol'dshtein and A Ukhlovmentioning

confidence: 99%

Weighted Sobolev spaces and embedding theorems

Gol’dshtein

Ukhlov

2009

Trans. Amer. Math. Soc.

165

133

View full text Add to dashboard Cite

show abstract

“…So, because Ω is a conformal regular domain, then for the function

g \in C^{1} (normalΩ)

the Poincaré inequality

∥ g - g_{normalΩ} ∣ L^{2} (normalΩ) ∥ \leq K^{*} ∥ g ∣ L^{1, 2} (normalΩ) ∥

holds with the exact constant

K^{*} = \sqrt{1 / λ_{2} [normalΩ]}

. Hence, using the “transfer” diagram we obtain

\begin{matrix} true \\ right ∥ f - f_{double-struckD, h} ∣ L^{2} (D, h) ∥ & = & left {((), \underset{double-struckD}{\int \int}, {| f (z) - f_{D, h} |}^{2}, h, (z), d, x, d, y)}^{\frac{1}{2}} \\ = & left {((), \underset{double-struckD}{\int \int}, {| f (z) - f_{D, h} |}^{2}, J, (z, φ), (z), d, x, d, y)}^{\frac{1}{2}} \\ = & left {((), \underset{normalΩ}{\int \int}, {| g (w) - g_{normalΩ} |}^{2}, d, u, d, w)}^{\frac{1}{2}} \\ \leq & left K^{*} (\underset{Ω}{-3pt \int -3pt \int}) \end{matrix}

…”

Section: The Weighted Eigenvalue Problemmentioning

confidence: 98%

Conformal spectral stability estimates for the Neumann Laplacian

Burenkov

Gol’dshtein

Ukhlov

2016

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

show abstract

“…So, for the function

g \in C_{0}^{\infty} (Ω)

the Poincaré inequality

‖g ∣ L^{2} (Ω)‖ \leq K^{*} ‖g ∣ L^{1, 2} (Ω)‖

holds with the exact constant

K^{*} = 1 / \sqrt{λ_{1} [Ω]}

. Hence, using the “transfer” diagram we obtain

\begin{matrix} true \\ right ‖f ∣ L^{2} (D, h)‖ & = & left {((), {\int \int}_{double-struckD}, {| f (z) |}^{2}, h, (z), d, x, d, y)}^{\frac{1}{2}} \\ = & left {((), {\int \int}_{double-struckD}, {| f (z) |}^{2}, J_{φ}, (z), d, x, d, y)}^{\frac{1}{2}} \\ = & left {((), {\int \int}_{Ω}, {(|| f \circ φ^{- 1} (w))}^{2}, d, u, d, v)}^{\frac{1}{2}} \\ \leq & left K^{*} {({-3pt \int -3pt \int}_{Ω} {|\nabla f \circ φ^{- 1}) (w)|}^{2} d u d v)}^{\frac{1}{2}} \\ = & left K^{*} () \end{matrix}

…”

Section: The Weighted Eigenvalue Problemmentioning

confidence: 99%

Conformal spectral stability estimates for the Dirichlet Laplacian

2015

Self Cite

View full text Add to dashboard Cite

We study the eigenvalue problem for the Dirichlet Laplacian in bounded simply connected plane domains ⊂ C by reducing it, using conformal transformations, to the weighted eigenvalue problem for the Dirichlet Laplacian in the unit disc D. This allows us to estimate the variation of the eigenvalues of the Dirichlet Laplacian upon domain perturbation via energy type integrals for a large class of "conformal regular" domains which includes all quasidiscs, i.e. images of the unit disc under quasiconformal homeomorphisms of the plane onto itself. Boundaries of such domains can have any Hausdorff dimension between one and two.

show abstract

Applications of change of variables operators for exact embedding theorems

Cited by 67 publications

References 4 publications

Weighted Sobolev spaces and embedding theorems

Weighted Sobolev spaces and embedding theorems

Conformal spectral stability estimates for the Neumann Laplacian

Conformal spectral stability estimates for the Dirichlet Laplacian

Contact Info

Product

Resources

About