Space quasiconformal composition operators with applications to Neumann eigenvalues

Spectral Stability Estimates of Neumann Divergence Form Elliptic Operators

We study spectral stability estimates of elliptic operators in divergence form $-\textrm{div} [A(w) \nabla g(w)]$ with the Neumann boundary condition in non-Lipschitz domains $Ω\subset \mathbb C$. The suggested method is based on connections of planar quasiconformal mappings with Sobolev spaces and its applications to the Poincaré inequalities.

Sharp bounds for composition with quasiconformal mappings in Sobolev spaces

Let $ϕ$ be a quasiconformal mapping, and let $T_ϕ$ be the composition operator which maps $f$ to $f\circϕ$. Since $ϕ$ may not be bi-Lipschitz, the composition operator need not map Sobolev spaces to themselves. The study begins with the behavior of $T_ϕ$ on $L^p$ and $W^{1,p}$ for $1<p<\infty$. This cases are well understood but alternative proofs of some known results are provided. Using interpolation techniques it is seen that compactly supported Bessel potential functions in $H^{s,p}$ are sent to $H^{s,q}$ whenever $0<s<1$ for appropriate values of $q$. The techniques used lead to sharp res…

The spectral estimates for the Neumann-Laplace operator in space domains

In this paper we prove discreteness of the spectrum of the Neu\-mann-Lap\-la\-ci\-an (the free membrane problem) in a large class of non-convex space domains. The lower estimates of the first non-trivial eigenvalue are obtained in terms of geometric characteristics of Sobolev mappings. The suggested approach is based on Poincaré-Sobolev inequalities that are obtained with the help of the composition operators theory for uniform Sobolev spaces. These composition operators are induced by a generalizations of conformal mappings that are mappings of bounded $2$-dilatation ($2$-quasiconformal mappi…

弱$p$-拟共形映射的精细化几何刻画

Refined geometric characterizations of weak $p$-quasiconformal mappings

本文研究弱$p$-拟共形映射的精细化几何刻画，证明几何$p$-伸缩在余集为$\sigma$-有限$(n-1)$-测度集的区域上有界的映射是$W^1_{p,\mathrm{loc}}$映射且在Sobolev空间上生成有界复合算子。

Composition Operators on Sobolev Spaces and Neumann Eigenvalues

In this paper we discuss applications of the geometric theory of composition operators on Sobolev spaces to the spectral theory of non-linear elliptic operators. The lower estimates of the first non-trivial Neumann eigenvalues of the $p$-Laplace operator in cusp domains $Ω\subset\mathbb R^n$, $n\geq 2$, are given.

Sobolev空间上复合算子的几何理论

Geometric theory of composition operators on Sobolev spaces

本文建立Sobolev空间上复合算子的几何理论框架，研究由拓扑映射诱导的有界嵌入算子，并将其应用于Sobolev嵌入定理、椭圆算子谱理论及连续介质力学。

On growth monotonicity estimates of the principal Dirichlet-Laplacian eigenvalue

In the present paper we obtain growth monotonicity estimates of the principal Dirichlet-Laplacian eigenvalue in bounded non-Lipschitz domains. The proposed method is based on composition operators generated by quasiconformal mappings and their applications to weighted Sobolev inequalities.

Estimates of Dirichlet eigenvalues of divergent elliptic operators in non-Lipschitz domains

We study spectral estimates of the divergence form uniform elliptic operators $-\textrm{div}[A(z) \nabla f(z)]$ with the Dirichlet boundary condition in bounded non-Lipschitz simply connected domains $Ω\subset \mathbb C$. The suggested method is based on the quasiconformal composition operators on Sobolev spaces with applications to the weighted Poincaré-Sobolev inequalities.

带权测度下外尖点区域中的非线性Neumann特征值

Nonlinear Neumann eigenvalues in outward cuspidal domains with weighted measure

该研究针对带权测度外尖点区域的非线性Neumann特征值问题，通过Sobolev空间复合算子建立到带权Lebesgue空间的嵌入，证明了谱问题可解性并给出特征值估计。

Sobolev $(p,q)$-扩展算子与Neumann特征值

Sobolev $(p,q)$-extension operators and Neumann eigenvalues

本文研究了Sobolev空间上的$(p,q)$-扩展算子，通过构造向外尖突域的扩展算子并估计其范数，证明了$p$-Laplace算子在向外尖突域中的非线性Neumann特征值的估计。

On the principal frequency of non-homogeneous membranes

We obtained estimates for first eigenvalues of the Dirichlet boundary value problem for elliptic operators in divergence form (i.e. for the principal frequency of non-homogeneous membranes) in bounded domains $Ω\subset \mathbb C$ satisfying quasihyperbolic boundary conditions. The suggested method is based on the quasiconformal composition operators on Sobolev spaces and their applications to constant estimates in the corresponding Sobolev-Poincaré inequalities. We also prove a variant of the Rayleigh-Faber-Khran inequality for a special case of these elliptic operators.

Spectral estimates of the $p$-Laplace Neumann operator in conformal regular domains

In this paper we study spectral estimates of the $p$-Laplace Neumann operator in conformal regular domains $Ω\subset\mathbb R^2$. This study is based on (weighted) Poincaré-Sobolev inequalities. The main technical tool is the composition operators theory in relation with the Brennan's conjecture. We prove that if the Brennan's conjecture holds then for any $p\in (4/3,2)$ and $r\in (1,p/(2-p))$ the weighted $(r,p)$-Poincare-Sobolev inequality holds with the constant depending on the conformal geometry of $Ω$. As a consequence we obtain classical Poincare-Sobolev inequalities and spectral estima…

关于向外Hölder尖角域中加权$p$-Laplace算子的$(p,q)$-特征值

On $(p,q)$-eigenvalues of the weighted $p$-Laplace operator in outward Hölder cuspidal domains

本文研究向外Hölder尖角域中加权$p$-Laplace算子的Neumann $(p,q)$-特征值问题，方法基于加权Sobolev空间上的复合算子。

The Fractional Lamé-Navier Operator: Appearances, Properties and Applications

We introduce and analyze an explicit formulation of fractional powers of the Lamé-Navier system of partial differential operators. We show that this fractional Lamé-Navier operator is a nonlocal integro-differential operator that appears in several widely-used continuum mechanics models. We demonstrate that the fractional Lamé-Navier operator can be obtained using compositions of nonlocal gradient operators. Additionally, the effective form of the fractional Lamé-Navier operator is the same as the operator obtained as a particular choice of parameters in state-based peridynamics. We further sh…