Related papers: Function spaces and extension results for nonlocal…
We study the Dirichlet problem for semilinear equations on general open sets with measure data on the right-hand side and irregular boundary data. For this purpose we develop the classical method of orthogonal projection. We treat in a…
In this work we provide a characterization of distinct type of (linear and non-linear) maps between Banach spaces in terms of the differentiability of certain class of Lipschitz functions. Our results are stated in an abstract bornological…
Dirichlet processes and their extensions have reached a great popularity in Bayesian nonparametric statistics. They have also been introduced for spatial and spatio-temporal data, as a tool to analyze and predict surfaces. A popular…
This paper introduces a notion of differential forms on closed, potentially fractal, subsets of Euclidean space by defining pointwise cotangent spaces using the restriction of $C^1$ functions to this set. Aspects of cohomology are…
This work is about global H\"older regularity for solutions to elliptic partial differential equations subject to mixed boundary conditions on irregular domains. There are two main results. In the first, we show that if the domain of the…
We show that every operator in $L^{2}$ has an associated measure on a space of functions and prove that it can be used to find solutions to abstract Cauchy problems, including partial differential equations. We find explicit formulas to…
In the paper is considered two problems on extension of operators whose range space for the first problem (or domain space for the second one) belongs to the fixed class of finite equivalence, which is generated by a given Banach space $X$.…
There are known trace and extension theorems relating functions in a weighted Sobolev space in a domain U to functions in a Besov space on the boundary bU. We extend these theorems to the case where the Sobolev exponent p is less than one…
Let $\mathscr T=(V, \mathcal E)$ be a leafless, locally finite rooted directed tree. We associate with $\mathscr T$ a one parameter family of Dirichlet spaces $\mathscr H_q~(q \geqslant 1)$, which turn out to be Hilbert spaces of…
We begin with an improvement to an extension result for subharmonic functions of Blanchet et al. With the aid of this improvement we then give extension results for subharmonic functions, for separately subharmonic functions, for harmonic…
We relate the (anisotropic) variable coefficient local and nonlocal Calder\'on problems by means of the Caffarelli-Silvestre extension. In particular, we prove that (partial) Dirichlet-to-Neumann data for the fractional Calder\'on problem…
An unsteady problem is considered for a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary…
The main result of this paper is a description of the space of functions on the unit circle, for which Krein's trace formula holds for arbitrary pairs of unitary operators with trace class difference. This space coincides with the space of…
We consider (locally) energy finite coordinates associated with a strongly local regular Dirichlet form on a metric measure space. We give coordinate formulas for substitutes of tangent spaces, for gradient and divergence operators and for…
Roughly speaking, functional analysis is the study of vector spaces of arbitrary dimension over the field of real or complex numbers, and the continuous linear mappings between such spaces. Naturally, the notion of continuity requires a…
Nonlocal problems for higher-order elliptic operators in dihedral and plane angles are considered. The Green formula is obtained, which leads to adjoint problems that take the form of nonlocal transmission problems in dihedral and plane…
We develop a framework for studying variational problems in Banach spaces with respect to gradient relations, which encompasses many of the notions of generalized gradients that appear in the literature. We stress the fact that our approach…
We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure…
In this paper, we provide a new means of establishing solvability of the Dirichlet problem on Lipschitz domains, with measurable data, for second order elliptic, non-symmetric divergence form operators. We show that a certain optimal…
In this paper we introduce new characterizations of spectral fractional Laplacian to incorporate nonhomogeneous Dirichlet and Neumann boundary conditions. The classical cases with homogeneous boundary conditions arise as a special case. We…