Generalization of Rellich-Kondrachov theorem and trace compacteness in the framework of irregular and fractal boundaries

On a Characterization of the Rellich-Kondrachov Theorem on Groups

Motivated by an eigenvalue-eigenfunction problem posed in IR^n x Ω, where Ω is a probability space, we are concerned in this paper with the Sobolev space on groups. Hence it is established an equivalence between locally compact Abelian groups and the space of solutions to the associated variational problem. Then, we study some conditions which characterize in a precisely manner the Rellich-Kondrachov Theorem, the principal ingredient to solve the variational problem.

Pseudodifferential Operators, Rellich-Kondrachov Theorem and Hardy-Sobolev Spaces

We will present versions of the Rellich-Kondrachov theorem for pseudo-differential operators acting on localizable Hardy spaces. One of the techniques includes boundedness properties for pseudodifferential operators with symbols in the Hörmander class, that might be of interest in his own right, extending results previously obtained by Goldberg, Alvarez and Hounie, Taylor - among others. As application, we obtain compact embedding results for distributions in nonhomogeneous localizable Hardy-Sobolev spaces.

Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d -sets

In the framework of the Laplacian transport, described by a Robin boundary value problem in an exterior domain in $\mathbb{R}^n$, we generalize the definition of the Poincaré-Steklov operator to $d$-set boundaries, $n-2< d<n$, and give its spectral properties to compare to the spectra of the interior domain and also of a truncated domain, considered as an approximation of the exterior case. The well-posedness of the Robin boundary value problems for the truncated and exterior domains is given in the general framework of $n$-sets. The results are obtained thanks to a generalization of the conti…

The solvability of Brezis-Nirenberg type problems of singular quasilinear elliptic equation

In this paper, we consider the existence and non-existence of non-trivial solution to a Brezis-Nirenberg type problem with singular weights. First, we obtain a compact imbedding theorem which is an extension of the classical Rellich-Kondrachov compact imbedding theorem, and consider the corresponding eigenvalue problem. Secondly, we deduce a Pohozaev type identity and obtained a non-existence result. Thirdly, based on a generalized concentration compactness principle, we will give some abstract conditions when the functional satisfies the (PS)$_c$ condition. Finally, based on the explicit form…

度量测度空间上的镜像流与$p$-Laplacian特征值问题

Mirror flows and $p$-Laplacian eigenvalue problems on metric measure spaces

本文研究镜像流作为Hilbertian梯度流的Banach空间对应物，在度量测度空间上的$p$-Laplacian特征值问题中发挥关键作用，建立了Ljusternik-Schnirelman型存在定理并证明Krasnoselskii谱中每个元素都是特征值。

带奇异权重的半空间上的高斯庞加莱不等式

Gaussian Poincaré inequalities on the half-space with singular weights

本文在半空间上带奇异权重的高斯测度下证明了 Rellich-Kondrachov 型定理和加权庞加莱不等式，并进一步研究了有界区域中测度为 $y^cdz$ 的局部情形。