Related papers: Gradient regularity for a class of elliptic obstac…

We provide comprehensive regularity results and optimal conditions for a general class of functionals involving Orlicz multi-phase of the type \begin{align} \label{abst:1} v\mapsto \int_{\Omega} F(x,v,Dv)\,dx, \end{align} exhibiting…

In this paper, we establish a regularity results for weak solutions of Robin problems driven by the well-known Orlicz $g$-Laplacian operator. Precisely, by using a suitable variation of the Moser iteration technique, we prove that every…

We analyze the local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian $(-\Delta)^s$ on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. For $1<p<2$, we obtain regularity in…

Minimizers of functionals of mixed local and nonlocal type are locally $C^{1,\beta}$-regular.

We prove that minimizers of the $L^{d}$-norm of the Hessian in the unit ball of $\mathbb{R}^d$ are locally of class $C^{1,\alpha}$. Our findings extend previous results on Hessian-dependent functionals to the borderline case and resonate…

This paper concerns almost minimizers of the functional $$ J(v,\Omega) = \int_\Omega \left( |D v^+|^p + |D v^-|^q \right) dx, $$ where $1<p \neq q< \infty$ and $\Omega$ is a bounded domain of $\mathbb{R}^n$, $n\geq 1$. We prove the…

We study energy functionals obtained by adding a possibly discontinuous potential to an interaction term modeled upon a Gagliardo-type fractional seminorm. We prove that minimizers of such non-differentiable functionals are locally bounded,…

In this paper we present a new proof of the sufficiency theorem for strong local minimizers concerning $C^1$-extremals at which the second variation is strictly positive. The results are presented in the quasiconvex setting, in accordance…

We consider elliptic operators with measurable coefficients and Robin boundary conditions on a bounded domain $\Omega \subset \mathbb{R}^d$ and show that the first eigenfunction $v$ satisfies $v(x) \ge \delta > 0$ for all $x \in…

It is establish regularity results for weak solutions of quasilinear elliptic problems driven by the well known $\Phi$-Laplacian operator given by \begin{equation*} \left\{\ \begin{array}{cl} \displaystyle-\Delta_\Phi u= g(x,u), &…

We prove the existence and $C^{1,\alpha}$ regularity of solutions to nonlocal fully nonlinear elliptic equations with gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be $C^1$ or…

Let $2\le n\le9$. Suppose that $f:R\to R$ is locally Lipschitz function satisfying $f(t)\ge A\min\{0,t\}-K$ for all $t\in R$ with some constant $A\ge0$ and $K\ge 0$. We establish an a priori interior H\"older regularity of $C^2$-stable…

We show that, for a fixed order $\gamma\geq 1$, each local minimizer of a rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary in the classical sense (corresponding to stationarity of order $1$),…

We consider autonomous integral functionals of the form $\mathcal F[u]:=\int_\Omega f(D u)\,dx$ with $u:\Omega\to\mathbb R^N$ $N\geq1$, where the convex integrand $f$ satisfies controlled $(p,q)$-growth conditions. We establish higher…

We pursue the study of a model convex functional with orthotropic structure and nonstandard growth conditions, this time focusing on the sub-quadratic case. We prove that bounded local minimizers are locally Lipschitz. No restriction on the…

In this paper, we establish an $\varepsilon$-regularity theorem for minimizers of an Alt-Phillips type functional subject to constraint maps. We prove that under sufficiently small energy, the minimizers exhibit regularity, and hence…

We prove higher regularity for nonlinear nonlocal equations with possibly discontinuous coefficients of VMO-type in fractional Sobolev spaces. While for corresponding local elliptic equations with VMO coefficients it is only possible to…

We prove global $W^{1,q}(\Omega,\mathbb{R}^m)$-regularity for minimisers of convex functionals of the form $\mathscr{F}(u)=\int_\Omega F(x,Du)\mathrm{d} x$. $W^{1,q}(\Omega,\mathbb{R}^m)$ regularity is also proven for minimisers of the…

We establish a partial $C^{1,\alpha}$ regularity result for minimizers of the optimal $p$-compliance problem with length penalization in any spatial dimension $N\geq 2$, extending some of the results obtained in…

We study the regularity of minimizers of a multiphase vectorial Bernoulli free boundary problem. This problem consists in a minimization problem for the Bernoulli functional over families of Sobolev functions with disjoint supports and non…