相关论文: On some equivalent norms in Sobolev spaces on boun…

A theory of Sobolev inequalities in arbitrary open sets of Euclidean space is established. Boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit…

In the setting of the Euclidean space equipped with an arbitrary Radon measure, we prove the equivalence between several notions of function of bounded variation present in the literature. We also study the relation between various…

We prove an estimate on the Hausdorff-dimension of the set of two-sided boundary points of general Sobolev-extension domains on Euclidean spaces. We also present examples showing lower bounds on possible dimension estimates of this type.

When a function belonging to a fractional-order Sobolev space is supported in a proper subset of the Lipschitz domain on which the Sobolev space is defined, how is its Sobolev norm as a function on the smaller set compared to its norm on…

A new notion of a Hausdorff-type operator on function spaces over domains in Euclidean spaces is introduced, and a sufficient condition for the boundedness of this operator on Sobolev spaces is proved. It is shown that this condition cannot…

The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients…

In the context of Euclidean spaces equipped with an arbitrary Radon measure, we prove the equivalence among several different notions of Sobolev space present in the literature and we characterise the minimal weak upper gradient of all…

For systems of ordinary differential equations on a compact interval, we study the character of solvability of the most general linear boundary-value problems in Sobolev spaces. We find the indices of these problems and obtain a criterion…

We establish equivalence between the boundedness of specific supremum operators and the optimality of function spaces in Sobolev embeddings acting on domains in ambient Euclidean space with a prescribed isoperimetric behavior. Our approach…

New isoperimetric inequalities for lower order eigenvalues of the Laplacian on closed hypersurfaces, of the biharmonic Steklov problems and of the Wentzell-Laplace on bounded domains in a Euclidean space are proven. Some open questions for…

We present various results on the equivalence and mapping properties under affine transformations of fractional-order Sobolev norms and semi-norms of orders between zero and one. Main results are mutual estimates of the three semi-norms of…

We prove a sharp logarithmic Sobolev inequality which holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature.

We deduce an extension theorem for the so-called Sobolev-Grand Lebesgue Spaces defined on the suitable subsets of the whole finite-dimensional Euclidean space, and estimate the norms of correspondent extension operator, which may be choosed…

The bounded variation seminorm and the Sobolev seminorm on compact manifolds are represented as a limit of fractional Sobolev seminorms. This establishes a characterization of functions of bounded variation and of Sobolev functions on…

In this paper we present a new characterization of Sobolev spaces on Euclidian spaces ($\mathbb{R}^n$). Our characterizing condition is obtained via a quadratic multiscale expression which exploits the particular symmetry properties of…

We consider Hardy operators on the half-space, that is, ordinary and fractional Schr\"odinger operators with potentials given by the appropriate power of the distance to the boundary. We show that the scales of homogeneous Sobolev spaces…

Using Fourier series representations of functions on axisymmetric domains, we find weighted Sobolev norms of the Fourier coefficients of a function that yield norms equivalent to the standard Sobolev norms of the function. This…

Sobolev embeddings, of arbitrary order, are considered into function spaces on domains of $\mathbb R^n$ endowed with measures whose decay on balls is dominated by a power $d$ of their radius. Norms in arbitrary rearrangement-invariant…

We investigate the comparability of generalized Triebel--Lizorkin and Sobolev seminorms on uniform and non-uniform sets when the integration domain is truncated according to the distance from the boundary. We provide numerous examples of…

Sobolev type inequalities involving homogeneous elliptic canceling differential operators and rearrangement-invariant norms on the Euclidean space are considered. They are characterized via considerably simpler one-dimensional Hardy type…