Related papers: Regularity results for solutions to obstacle probl…
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…
Regularity properties of solutions to variational problems are established for a broad class of strictly convex splitting-type energy densities of the principal form $f$: $\mathbb{R}^2 \to \mathbb{R}$, \[ f(\xi_1,\xi_2) = f_1\big( \xi_1…
In this paper, we consider a new class of multi phase operators with variable exponents, which reflects the inhomogeneous characteristics of hardness changes when multiple different materials are combined together. We at first deal with the…
We establish the higher fractional differentiability of bounded minimizers to a class of obstacle problems with non-standard growth conditions of the form \begin{gather*} \min \biggl\{ \displaystyle\int_{\Omega} F(x,Dw)dx \ : \ w \in…
We present recent advances in the regularity theory for weak solutions to some classes of elliptic and parabolic equations with strongly singular or degenerate structure. The equations under consideration satisfy standard $p$-growth and…
This paper studies the Sobolev regularity estimates of weak solutions of a class of singular quasi-linear elliptic problems of the form $u_t - \mbox{div}[\mathbb{A}(x,t,u,\nabla u)]= \mbox{div}[{\mathbf F}]$ with homogeneous Dirichlet…
We consider elliptic problems with nonclassical boundary conditions that contain additional unknown functions on the border of the domain of the elliptic equation and also contain boundary operators of higher orders with respect to the…
We introduce the space of rough paths with Sobolev regularity and the corresponding concept of controlled Sobolev paths. Based on these notions, we study rough path integration and rough differential equations. As main result, we prove that…
Despite significant recent advances in the regularity theory for obstacle problems with integro-differential operators, some fundamental questions remained open. On the one hand, there was a lack of understanding of parabolic problems with…
Optimal second-order regularity in the space variables is established for solutions to Cauchy-Dirichlet problems for nonlinear parabolic equations and systems of $p$-Laplacian type, with square-integrable right-hand sides and initial data…
The aim of the paper is to show that the solutions to variational problems with non-standard growth conditions satisfy a corresponding variational inequality without any smallness assumptions on the gap between growth and coercitivity…
In this article we discuss quantitative properties of convex integration solutions arising in problems modeling shape-memory materials. For a two-dimensional, geometrically linearized model case, the hexagonal-to-rhombic phase…
In the present paper, we establish sharp Sobolev estimates for solutions of fully nonlinear parabolic equations, under minimal, asymptotic, assumptions on the governing operator. In particular, we prove that solutions are in…
In this article, first we address the regularity of weak solution for a class of $p$-fractional Choquard equations: \begin{equation*} \;\;\; \left.\begin{array}{rl}…
We obtain some regularity results for solutions to vectorial $p$-Laplace equations $$ -{\boldsymbol \Delta}_p{\boldsymbol u}=-\operatorname{\bf div}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x,{\boldsymbol u})\,\, \mbox{…
This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations extending our findings in [22] to domains of polyhedral type. In particular, we study the smoothness in the specific scale…
We prove an a priori estimate for the second derivatives of local minimizers of integral functionals of calculus of variation with convex integrand with respect to the gradient variable, assuming that the function that measures the…
We prove that a local, weak Sobolev inequality implies a global Sobolev estimate using existence and regularity results for a family of $p$-Laplacian equations. Given $\Omega\subset\mathbb{R}^n$, let $\rho$ be a quasi-metric on $\Omega$,…
We refine the theory of the cohomological equation for translation flows on higher genus surfaces with the goal of proving optimal results on the Sobolev regularity of solutions and of distributional obstructions. For typical translation…
We develop a systematic study of the interior Sobolev regularity of weak solutions to the mixed local and nonlocal $p$-Laplace equations. To be precise, we show that the weak solution $u$ belongs to $W^{2, p}_\mathrm{loc}$ and even $W^{2,…