Related papers: On the Two Obstacles Problem in Orlicz-Sobolev Spa…

We investigate partial regularity for vector valued local minimizers of double phase functionals, under vectorial obstacle type constraints satisfying appropriate topological properties.

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…

Double forms are sections of the vector bundles $\Lambda^{k}T^*\mathcal{M}\otimes \Lambda^{m}T^*\mathcal{M}$, where in this work $(\mathcal{M},\mathfrak{g})$ is a compact Riemannian manifold with boundary. We study graded second-order…

A classical regularity result is that non-negative solutions to the Dirichlet problem $\Delta u =f$ in a bounded domain $\Omega$, where $f\in L^q(\Omega)$, $q>\frac{n}2$, satisfy $\|u\|_{L^\infty(\Omega)} \leq C\|f\|_{L^q(\Omega)}$. We…

We show that any minimizer of the well-known ACF functional (for the $p$-Laplacian) is a viscosity solution. This allows us to establish a uniform flatness decay at the two-phase free boundary points to improve the flatness, that boils down…

We provide a direct proof of existence and uniqueness of weak solutions to a broad family of strongly nonlinear elliptic equations with lower order terms. The leading part of the operator satisfies general growth conditions settling the…

In this paper, we consider the following problem involving fractional Laplacian operator: \begin{equation}\label{eq:0.1} (-\Delta)^{\alpha} u= |u|^{2^*_\alpha-2-\varepsilon}u + \lambda u\,\, {\rm in}\,\, \Omega,\quad u=0 \,\, {\rm on}\, \,…

For the parabolic obstacle-problem-like equation $$\Delta u - \partial_t u = \lambda_+ \chi_{\{u>0\}} - \lambda_- \chi_{\{u<0\}} ,$$ where $\lambda_+$ and $\lambda_-$ are positive Lipschitz functions, we prove in arbitrary finite dimension…

We study the boundary regularity of solutions of elliptic operators in divergence form with $C^{0,\alpha}$ coefficients or operators which are small perturbations of the Laplacian in non-smooth domains. We show that, as in the case of the…

We consider a stochastic parabolic partial differential equation with Dirichlet boundary conditions, multiplicative stochastic noise, and a monotone parabolic operator A. The growth and coercivity of A is controlled by a general N-function…

We consider the Cauchy problem for non-autonomous forms inducing elliptic operators in divergence form with Dirichlet, Neumann, or mixed boundary conditions on an open subset $\Omega$ $\subseteq$ R n. We obtain maximal regularity in L 2…

Let $\Omega \subset \mathbb{R}^{n+1}$ be a bounded chord-arc domain, let $\mathcal L=-{\rm div} A\nabla$ be an elliptic operator in $\Omega$ associated with a matrix $A$ having Dini mean oscillation coefficients, and let $1<p\leq 2$. In…

In this paper, we study the existence of non-negative non-trivial solutions for a class of double-phase problems where the source term is a Caratheodory function that satisfies the Ambrosetti-Rabinowitz type condition in the framework of…

It is established existence, uniqueness and multiplicity of solutions for a quasilinear elliptic problem problems driven by $\Phi$-Laplacian operator. Here we consider the reflexive and nonreflexive cases using an auxiliary problem. In…

In this paper we establish optimal regularity estimates and smoothness of free boundaries for nonlocal obstacle problems governed by a very general class of integro-differential operators with possibly singular kernels. More precisely, in…

This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable t-independent coefficients in spaces of fractional smoothness, in Besov and weighted Sobolev classes. We establish…

We study the obstacle problem for the Evolutionary p-Laplace Equation when the obstacle is discontinuous and without regularity in the time variable. Two quite different procedures yield the same solution.

We investigate two related problems concerning the dimension of joint eigenspaces of the Laplace--Beltrami operator and a finite set of Hecke operators on $\mathbb{X}=\mathrm{PGL}_2(\mathbb{Z})\backslash \mathbb{H}$. First, we consider…

We study thin obstacle problems involving the energy functional with $p(x)$-growth. We prove higher integrability and H\"{o}lder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable…

In this article we study solutions to the (interior) thin obstacle problem under low regularity assumptions on the coefficients, the obstacle and the underlying manifold. Combining the linearization method of Andersson \cite{An16} and the…