Related papers: $\Gamma$-convergence for power-law functionals wit…

Let $\Gamma$ be a rectifiable Jordan curve in the complex plane, $\Omega_i$ and $\Omega_e$ respectively the interior and exterior domains of $\Gamma$, and $p\geq 2$. Let $E$ be the vector space of functions defined on $\Gamma$ consisting of…

The function $\Gamma$ on the space of graphons, introduced in [CGH$^+$15], aims to measure the extent to which a graphon $w$ exhibits the Robinson property: for all $x<y<z$, $w(x,z)\leq \min\{ w(x,y),w(y,z)\}$. Robinson graphons form a…

We are concerned with Dirichlet problems of the form $${\mathop{\rm div}\nolimits} (|D u|^{p-2}Du)+f(u)=0\ \mbox{ in }\Omega,\qquad u=0\ \mbox{ on }\partial\Omega, $$ where $\Omega$ is a bounded domain of $\mathbb{R}^n$, $n\ge 2$, $1<p<n$…

We prove {the first} regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain $\Omega$ is obtained as the integral of a cost function $j(u,x)$…

Fixing an arithmetic lattice $\Gamma$ in an algebraic group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $\Delta$ with $[\Gamma : \Gamma \cap \Delta] [\Delta: \Gamma \cap \Delta] =…

In view of the applications to the asymptotic analysis of a family of obstacle problems, we consider a class of convex local functionals $F(u,A)$, defined for all functions $u$ in a suitable vector valued Sobolev space and for all open sets…

Let $\mathcal{U(\alpha, \lambda)}$, $0<\alpha <1$, $0 < \lambda <1$ be the class of functions $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ satisfying $$\left|\left(\frac{z}{f(z)}\right)^{1+\alpha}f'(z)-1\right|<\lambda$$ in the unit disc ${\mathbb…

It is shown that if $p_n$ is a sequence of continuous, unbounded exponents on a bounded, smooth domain $\Omega\subset {\mathbb R}^n$ with $1<\inf\limits_{x\in \Omega}p_n(x)$ and $p_n\rightarrow \infty$ uniformly, then the sequence $(u_n)$…

A class of subharmonic functions represented by the modified kernels are proved to have the growth estimates u(x) =o(x_{n}^{1-alpha}|x|^{m+alpha})at infinity in the upper half space of Rn, which generalizes the growth properties of analytic…

We study the integral representation of $\Gamma$-limits of $p$-coercive integral functionals of the calculus of variations in the spirit of \cite{dalmaso-modica86}. We use infima of local Dirichlet problems to characterize the limit…

Let $X$ be a rearrangement-invariant space over a non-atomic $\sigma$-finite measure space $(\mathscr{R},\mu)$ and let $\alpha\in(0,\infty)$. We define the functional \begin{equation*} \|f\|_{X^{\langle \alpha \rangle}} =…

The aim of this paper is investigating the existence of one or more critical points of a family of functionals which generalizes the model problem \[ \bar J(u)\ =\ \frac1p\ \int_\Omega \bar A(x,u)|\nabla u|^p dx - \int_\Omega G(x,u) dx \]…

We establish local higher integrability and differentiability results for minimizers of variational integrals $$ \mathfrak{F}(v,\Omega) = \int_{\Omega} /! F(Dv(x)) \, dx $$ over $W^{1,p}$--Sobolev mappings $u \colon \Omega \subset {\mathbb…

If $u : \Omega\subset \mathbb{R}^d \to {\rm X}$ is a harmonic map valued in a metric space ${\rm X}$ and ${\sf E} : {\rm X} \to \mathbb{R}$ is a convex function, in the sense that it generates an ${\rm EVI}_0$-gradient flow, we prove that…

In this article we obtain concentration inequalities for Poisson $U$-statistics $F_m(f,\eta)$ of order $m\ge 1$ with kernels $f$ under general assumptions on $f$ and the intensity measure $\gamma \Lambda$ of underlying Poisson point process…

We present an extension of the classical De Giorgi class, and then we show that functions in this new class are locally bounded and locally H\"older continuous. Some applications are given. As a first application, we give a regularity…

Here is one of the results obtained in this paper: Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain, let $q>1$, with $q<{{n+2}\over {n-2}}$ if $n\geq 3$ and let $\lambda_1$ be the first eigenvalue of the problem $$\cases{-\Delta…

We prove that if $(u,\Gamma)$ is a minimizer of the functional $$ J(u,\Gamma)=\int_{B_1(0)\setminus \Gamma}|\nabla u|^2dx +\H^1(\Gamma) $$ and $\Gamma$ connects $\partial B_1(0)$ to a point in the interior, then $\Gamma$ satisfies a…

In this work, we revisit the following estimate due to Dahlberg \cite{Dahl}. Let $\textit{\textbf x}_0$ a fixed point in a bounded Lipschitz domain $\Omega$. Then there exists a constant $C > 0$ such that if $u$ is a harmonic function in…

The aim of this paper is to deal with the asymptotics of generalized Orlicz norms when the lower growth rate tends to infinity. $\Gamma$-convergence results and related representation theorems in terms of $L^\infty$ functionals are proven…