Related papers: Local compactness and nonvanishing for weakly sing…

We consider the Kato and the Dynkin class and their local counterparts on a smooth Riemannian manifold as Fr\'{e}chet spaces. Based on recent results by Carron, Mondello and Tewodrose we show that for a Riemannian manifold $(X,g)$ of…

This paper constructs a unified family of nonconforming finite element spaces for $H\Lambda^k$ in $\mathbb{R}^n$ ($0\leqslant k\leqslant n$, $n\geqslant 1$). The spaces employ piecewise Whitney forms as shape functions, and include the…

We study nonnegative optimizers of a Gagliardo-Nirenberg type inequality $$\iint_{\mathbb{R}^N \times \mathbb{R}^N} \frac{|u(x)|^p\,|u(y)|^p}{|x - y|^{N-\alpha}} dx\, dy\le C\Big(\int_{{\mathbb R}^N}|u|^2 dx\Big)^{p\theta}…

In this article, under mild constraints on the sectional curvature, we exploit a divergence formula for symmetric endomorphisms to deduce a general Poincar\'e type inequality. We apply such inequality to higher-order mean curvature of…

We study a family of non-convex functionals $\{\mathcal{E}\}$ on the space of measurable functions$u: \Omega_1\times\Omega_2 \subset \mathbb{R}^{n_1}\times\mathbb{R}^{n_2} \to \mathbb{R}$. These functionals vanish on the non-convex subset…

Let $d \ge 1$, $p \ge d$, and let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^d$. We prove some exponential integrability in the spirit of Moser-Trudinger's inequalities for measurable functions $u$ defined in $\Omega$ such that…

Let $ \mathcal D\equiv G/K $ be an irreducible bounded symmetric domain. Using a vector-valued version of Mui\'c's integral non-vanishing criterion for Poincar\'e series on locally compact Hausdorff groups, we study the non-vanishing of…

We study local regularity properties for solutions of linear, non-uniformly elliptic equations. Assuming certain integrability conditions on the coefficient field, we prove local boundedness and Harnack inequality. The assumed integrability…

We prove a general result about the behaviour of minimizing sequences for nonlocal shape functionals satisfying suitable structural assumptions. Typical examples include functions of the eigenvalues of the fractional Laplacian under…

In this paper, we study the optimal constant in the nonlocal nonlinear Poincar\'e-Wirtinger inequality in $(a,b)\subset\mathbb R$: \begin{equation*} \lambda_\alpha(p,q,r){\left(\int_{a}^{b}|u|^{q}dx\right)^\frac…

In this paper we consider a very general form of a non-local energy in integral form, which covers most of the usual ones (for instance, the sum of a positive and a negative power). Instead of admitting only sets, or $L^\infty$ functions,…

We prove a compact embedding theorem in a class of spaces of piecewise H1 functions subordinated to a class of shape regular, but not necessarily quasi-uniform triangulations of a polygonal domain. This result generalizes the…

We prove the existence of periodic tessellations of $\mathbb{R}^N$ minimizing a general nonlocal perimeter functional, defined as the interaction between a set and its complement through a nonnegative kernel, which we assume to be either…

We prove a weak-strong convergence result for functionals of the form $\int_{\mathbb{R}^N} j(x, u, Du)\,dx$ on $W^{1,p}$, along equiintegrable sequences. We will then use it to study cases of equality in the extended Polya-Szeg\"o…

In this paper, we prove a Poincar\'e-type inequality for any set of finite perimeter which is stable with respect to the free energy among volume-preserving perturbation, provided that the Hausdorff dimension of its singular set is at most…

We prove local boundedness of local minimizers of scalar integral functionals $\int_\Omega f(x,\nabla u(x))\,dx$, $\Omega\subset\mathbb R^n$ where the integrand satisfies $(p,q)$-growth of the form \begin{equation*} |z|^p\lesssim…

We initiate a systematic study of intrinsic dimensional versions of classical functional inequalities which capture refined properties of the underlying objects. We focus on model spaces: Euclidean space, Hamming cube, and manifolds of…

This paper investigates the preservation of local minimizers and strong minimizers of extended-real-valued lower semicontinuous functions under taking their Moreau envelopes. We address a general setting of Banach spaces, while all the…

We study regularity properties for solutions to the nakedly degenerate elliptic equation $a_{ij}\partial_{ij}u =0$, where the coefficients satisfy $I \ge a_{ij}(x) \ge \lambda(x) I$ and the only assumption is that $\lambda^{-1} \in L^p$. We…

The purpose of this paper is to study the lower semicontinuity with respect to the strong $L^1$-convergence, of some integral functionals defined in the space SBD of special functions with bounded deformation. Precisely, let $U$ be a…