Related papers: Regularity in gradient constraint problem and thin…

We investigate an overdetermined Torsion problem, with a non-constant positively homogeneous boundary constraint on the gradient. We interpret this problem as the Euler equation of a shape optimization problems, we prove existence and…

In this article we study optimization problems ruled by $\alpha$-fractional diffusion operators with volume constraints. By means of penalization techniques we prove existence of solutions. We also show that every solution is locally of…

This paper provides theoretical consistency results for compressed modes. We prove that as L1 regularization term in certain non-convex variational optimization problems vanishes, the solutions of the optimization problem and the…

Li and Vogelius, and Li and Nirenberg obtained piecewise $C^{1,\gamma}$-regularity for linear elliptic problems with piecewise $C^{\gamma}$-coefficients which come from composite materials. In this paper, we obtain piecewise…

We introduce and study co-dimension one area-minimizing locally rectifiable currents $T$ with $C^{1,\alpha}$ tangentially immersed boundary: $\partial T$ is locally a finite sum of orientable co-dimension two submanifolds which only…

We establish the optimal $C_{H}^{1,1}$ interior regularity of solutions to \[ \Delta_{H}u=f\chi_{\{u\ne0\}}, \] where $\Delta_{H}$ denotes the sub-Laplacian operator in a stratified group. We assume the weakest regularity condition on $f$,…

We consider a wide class of fully nonlinear integro-differential equations that degenerate when the gradient of the solution vanishes. By using compactness and perturbation arguments, we give a complete characterization of the regularity of…

In these notes we prove log-type stability for the Calder\'on problem with conductivities in $ C^{1,\varepsilon}(\bar{\Omega}) $. We follow the lines of a recent work by Haberman and Tataru in which they prove uniqueness for $…

We study $n$-dimensional area-minimizing currents $T$ in $\mathbb{R}^{n+1},$ with boundary $\partial T$ satisfying two properties: $\partial T$ is locally a finite sum of $(n-1)$-dimensional $C^{1,\alpha}$ orientable submanifolds which only…

We study the regularity of optimal transport maps between convex domains with quadratic cost. For nondegenerate $C^{\alpha}$-densities, we prove $C^{1, 1-\varepsilon}$-regularity of the potentials up to the boundary. If in addition the…

We revisit the classical theory of linear second-order uniformly elliptic equations in divergence form whose solutions have H\"older continuous gradients, and prove versions of the generalized maximum principle, the $C^{1,\alpha}$-estimate,…

We study the boundary regularity of solutions of elliptic operators in divergence form with $C^{0,\alpha}$ coefficients or operators which are small perturbations of the Laplacian in non-smooth domains. We show that, as in the case of the…

In this paper, we establish global $C^{1,\alpha}$-regularity for bounded generalized solutions of elliptic equations in divergence form with Musielak-Orlicz growth and subject to Dirichlet or Neumann boundary conditions. In fact, our…

We study regularity properties for solutions to elliptic equations that are degenerate or singular along orthogonal hyperplanes. The degenerate ellipticity is carried out by a weight term which is the monomial product of different powers of…

We extend classical results on variational inequalities with convex sets with gradient constraint to a new class of fractional partial differential equations in a bounded domain with constraint on the distributional Riesz fractional…

We prove that a stable minimal hypersurface of an open ball having a singular set of locally finite codimension 2 Hausdorff measure which is weakly close to a multiplicity 2 hyperplane is a 2-valued C^{1, alpha} graph in the interior.…

We prove an Allard-type regularity theorem for free-boundary minimal surfaces in Lipschitz domains locally modelled on convex polyhedra. We show that if such a minimal surface is sufficiently close to an appropriate free-boundary plane,…

We prove an $L^p(I,C^\alpha(\Omega))$ regularity result for a reaction-diffusion equation with mixed boundary conditions, symmetric $L^\infty$ coefficients and an $L^\infty$ initial condition. We provide explicit control of the…

In this paper, we are interested in shape optimization problems involving the ge ometry (normal, curvatures) of the surfaces. We consider a class of hypersurface s in $\mathbb{R}^{n}$ satisfying a uniform ball condition and we prove the…

In this work, we study regularity properties for nonvariational singular elliptic equations ruled by the infinity Laplacian. We obtain optimal $C^{1,\alpha}$ regularity along the free boundary. We also show existence of solutions,…