Related papers: Sobolev regularity of occupation measures and path…

We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds…

Let (X, \r{ho},\mu) be a space of homogeneous type, a variable exponent satisfying the globally log-Holder continuous condition. In this article, the author introduce the variable fractional Sobolev spaces on X via Haj{\l}asz gradient.…

Consider a Lipschitz domain $\Omega$ and a measurable function $\mu$ supported in $\overline\Omega$ with $\left\|{\mu}\right\|_{L^\infty}<1$. Then the derivatives of a quasiconformal solution of the Beltrami equation $\overline{\partial} f…

The aim of the paper is to develop a general theory of solvability of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of arbitrary order in Sobolev spaces. Boundary conditions are allowed to be…

This paper provides a comprehensive Sobolev regularity theory for the Dirichlet problem of stochastic partial differential equations in $C^{1,\sigma}$ open sets. We consider substantially large classes of nonlocal operators and generalized…

We prove quasi-monotonicity formulae for classical obstacle-type problems with quadratic energies with coefficients in fractional Sobolev spaces, and a linear term with a Dini-type continuity property. These formulae are used to obtain the…

In a Banach space $X$ endowed with a nondegenerate Gaussian measure, we consider Sobolev spaces of real functions defined in a sublevel set $O= \{x\in X:\;G(x) <0\}$ of a Sobolev nondegenerate function $G:X\mapsto \R$. We define the traces…

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities.…

In this work, we establish some abstract results on the perspective of the fractional Musielak-Sobolev spaces, such as: uniform convexity, Radon-Riesz property with respect to the modular function, $(S_{+})$-property, Brezis-Lieb type Lemma…

First of all, we establish compactness of continuous mappings of the Orlicz--Sobolev classes $W^{1,\varphi}_{\rm loc}$ with the Calderon type condition on $\varphi$ and, in particular, of the Sobolev classes $W^{1,p}_{\rm loc}$ for $p>n-1$…

In this paper, by using a characterization of functions having fractional derivative, we propose a rigorous fractional Lyapunov function candidate method to analyze stability of fractional-order nonlinear systems. First, we prove an…

The paper contains a review of results on linear systems of ordinary differential equations of an arbitrary order on a finite interval with the most general inhomogeneous boundary conditions in Sobolev spaces. The character of the…

In the past years, the phenomenon of fractional regularity has been addressed for a large class of linear and/or quasilinear differential operators, mostly, in terms of certain Besov spaces. As it turned out, for equations governed by the…

We obtain an improved Sobolev inequality in H^s spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimizers and the compactness up to symmetry of optimizing sequences for the usual Sobolev embedding.…

In this article, we introduce the notion of the Riemann-Liouville fractional integral of set-valued mappings via integrable selections. We establish fundamental properties of this fractional integral, including convexity, boundedness, and…

This paper presents three new families of fractional Sobolev spaces and their accompanying theory in one-dimension. The new construction and theory are based on a newly developed notion of weak fractional derivatives, which are natural…

In this paper, we investigate the existence of weak solution for a fractional type problems driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions. We first extend…

We study the interplay between the regularity of paths and Hamiltonians in the theory of pathwise Hamilton-Jacobi equations with the use of interpolation methods. The regularity of the paths is measured with respect to Sobolev, Besov,…

We prove some regularity results for a priori bounded local minimizers of non-autonomous integral functionals of the form $$\mathcal{F}(v,\Omega)=\int_\Omega F(x,Dv)dx,$$ under the constraint $v \ge \psi$ a.e. in $\Omega$, where $\psi$ is a…

This study presents new estimates for fractional derivatives without singular kernels defined by some specific functions. Based on obtained inequalities, we give a useful method to establish the global stability of steady states for…