相关论文: Removable sets in elliptic equations with Musielak…

We analyze fine properties of solutions to quasilinear elliptic equations with double phase structure and characterize, in the terms of intrinsic Hausdorff measures, the removable sets for H\"older continuous solutions.

We study nonlinear measure data elliptic problems involving the operator exposing generalized Orlicz growth. Our framework embraces reflexive Orlicz spaces, as well as natural variants of variable exponent and double-phase spaces.…

We characterize removable sets for H\"older continuous solutions to degenerate parabolic equations of $p$-growth. A sufficient and necessary condition for a set to be removable is given in terms of an intrinsic parabolic Hausdorff measure,…

Anisotropic elliptic equations of the second order with variable exponents in nonlinearities and the right-hand side as a diffuse measure are considered in the space $\mathbb{R}^n$. The existence of an entropy solution in anisotropic…

This paper investigates elliptic obstacle problems with generalized Orlicz growth involving measure data, which includes Orlicz growth, variable exponent growth, and double-phase growth as specific cases of this setting. First, we establish…

In this article, we consider a higher-order elliptic equation with nonsmooth coefficients with respect to Orlicz spaces on the domain $\Omega\subset\mathbb{R}^{n}$. The separable subspace of this space is distinguished in which infinitely…

We study the subsets of metric spaces that are negligible for the infimal length of connecting curves; such sets are called metrically removable. In particular, we show that every totally disconnected set with finite Hausdorff measure of…

We investigate how the integrability of the derivatives of Orlicz-Sobolev mappings defined on open subsets of $\mathbb{R}^n$ affect the sizes of the images of sets of Hausdorff dimension less than $n$. We measure the sizes of the image sets…

In this note we give an upper bound on the Hausdorff dimension of removable setsfor elliptic and canceling homogeneous differential operators with constant coefficients in the class of bounded functions, using a simple extension of…

Boundary value problems for a class of quasilinear elliptic equations, with an Orlicz type growth and L^1 right-hand side are considered. Both Dirichlet and Neumann problems are contemplated. Existence and uniqueness of generalized…

We introduce a new notion of a harmonic measure for a $d$-dimensional set in $\R^n$ with $d<n-1$, that is, when the codimension is strictly bigger than 1. Our measure is associated to a degenerate elliptic PDE, it gives rise to a…

In this paper we study critical sets of solutions $u_\e$ of second-order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. We show that the $(d-2)$-dimensional Hausdorff measures of the critical sets…

We consider the problem of minimizing variational integrals defined on \cc{nonlinear} Sobolev spaces of competitors taking values into the sphere. The main novelty is that the underlying energy features a non-uniformly elliptic integrand…

We explore quantitative propagation of smallness for solutions of two-dimensional elliptic equations from sets of positive $\delta$-dimensional Hausdorff content for any $\delta>0$. In particular, the gradients of solutions to divergence…

We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consider \begin{equation*} -{\rm div} A(x,\nabla u)= f\in L^1(\Omega), \end{equation*} on…

We study solutions to measure data elliptic systems with Uhlenbeck-type structure that involve operator of divergence form, depending continuously on the spacial variable, and exposing doubling Orlicz growth with respect to the second…

It is well known that a purely unrectifiable set cannot support a harmonic measure which is absolutely continuous with respect to the Hausdorff measure of this set. We show that nonetheless there exist elliptic operators on (purely…

We study the gradient regularity of solutions to measure data elliptic systems with Uhlenbeck-type structure and Orlicz growth. For any bounded Borel measure, pointwise estimates for the gradient of solutions are provided in terms of the…

We investigate solutions to nonlinear elliptic Dirichlet problems of the type \[ \left\{\begin{array}{cl} - {\rm div} A(x,u,\nabla u)= \mu &\qquad \mathrm{ in}\qquad \Omega, u=0 &\qquad \mathrm{ on}\qquad \partial\Omega, \end{array}\right.…

We study the removability of a singular set for elliptic equations involving weight functions and variable exponents. We consider the case where the singular set satisfies conditions related to some generalization of upper Minkowski content…