Related papers: Lipschitz bounds for integral functionals with $(p…

We study the minimization of convex, variational integrals of linear growth among all functions in the Sobolev space $W^{1,1}$ with prescribed boundary values (or its equivalent formulation as a boundary value problem for a degenerately…

We consider functionals of the form $$\mathcal{F}(u):=\int_\Omega\!F(x,u,\nabla u)\,\mathrm{d} x,$$ where $\Omega\subseteq\mathbb{R}^n$ is open and bounded. The integrand $F\colon\Omega\times\mathbb{R}^N\times\mathbb{R}^{N\times…

We prove global $W^{1,q}(\Omega,\mathbb{R}^N)$-regularity for minimisers of $\mathscr{F}(u)=\int_\Omega F(x,\mathrm{D}u)\mathrm{d} x$ satisfying $u\geq \psi$ for a given Sobolev obstacle $\psi$. $W^{1,q}(\Omega,\mathbb{R}^m)$ regularity is…

We establish maximal local regularity results of weak solutions or local minimizers of \[ \operatorname{div} A(x, Du)=0 \quad\text{and}\quad \min_u \int_\Omega F(x,Du)\,dx, \] providing new ellipticity and continuity assumptions on $A$ or…

We prove higher integrability for local minimizers of the double-phase orthotropic functional \[ \sum_{i=1}^{n}\int_\Omega\left(\left|u_{x_i}\right|^p+a(x)\left| u_{x_i}\right|^q\right)dx \] when the weight function $a \geq0$ is assumed to…

We consider a class of {energy integrals}, associated to nonlinear and non-uniformly elliptic equations, with integrands $f(x,u,\xi)$ satisfying anisotropic $p_i,q$-growth conditions of the form $$ \sum_{i=1}^n \lambda_i (x)|\xi_i|^{p_i}\le…

We study robust regularity estimates for local minimizers of nonlocal functionals with non-standard growth of $(p,q)$-type and for weak solutions to a related class of nonlocal equations. The main results of this paper are local boundedness…

We consider some energy integrals under slow growth and we prove that the local minimizers are locally Lipschitz continuous. Many examples are given, either with subquadratic $p,q-$growth and/or anisotropic growth.

This article deals with the Lipschitz regularity of the ''approximate`` minimizers for the Bolza type control functional of the form \[J_t(y,u):=\int_t^T\Lambda(s,y(s), u(s))\,ds+g(y(T))\] among the pairs $(y,u)$ satisfying a prescribed…

We deal with homogeneous Dirichlet and Neumann boundary-value problems for anisotropic elliptic operators of p-Laplace type. They emerge as Euler-Lagrange equations of integral functionals of the Calculus of Variations built upon possibly…

We prove higher differentiability of bounded local minimizers to some widely degenerate functionals, verifying superquadratic anisotropic growth conditions. In the two dimensional case, we prove that local minimizers to a model functional…

We prove that, given~$p>\max\left\{\frac{2n}{n+2},1\right\}$, the nonnegative almost minimizers of the nonlinear free boundary functional $$ J_p(u,\Omega):=\int_{\Omega}\Big( |\nabla u(x)|^p+\chi_{\{u>0\}}(x)\Big)\,dx$$ are Lipschitz…

In this paper we are concerned with the regularity of solutions to a nonlinear elliptic system of $m$ equations in divergence form, satisfying $p$ growth from below and $q$ growth from above, with $p \leq q$; this case is known as $p,…

We establish local regularity results for minimizers of autonomous vectorial integrals of Calculus of Variations, assuming $\psi$-growth conditions and imposing $\varphi$-quasiconvexity only in an asymptotic sense, both in the sub-quadratic…

We prove higher summability for the gradient of minimizers of strongly convex integral functionals of the Calculus of Variations with (p,q)-Growth conditions in low dimension. Our procedure is set in the framework of Fractional Sobolev…

We start presenting an $L^{\infty}$-gradient bound for solutions to non-homogeneous $p$-Laplacean type systems and equations, via suitable non-linear potentials of the right hand side. Such a bound implies a Lorentz space characterization…

We establish partial regularity results for minimizers of a class of functionals depending on differential expressions based on elliptic operators. Specifically, we focus on functionals of Orlicz growth with a natural strong quasiconvexity…

We prove partial regularity for minimizers of quasiconvex functionals of the type $\int_\Omega f(x,Du) dx$ with $p(x)$ growth with respect to the second variable. The proof is direct and uses a method of $A$-harmonic approximation.

We prove global Lipschitz regularity for a wide class of convex variational integrals among all functions in $W^{1,1}$ with prescribed (sufficiently regular) boundary values, which are not assumed to satisfy any geometrical constraint (as…

We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional $$\label{E} E(u,\Omega) = \int_\Omega |\nabla u|^2 dX + \mathcal{H}^n(\{u>0\} \cap \{x_{n+1} = 0\}), \quad…