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L. A. Caffarelli

Publications and source records attributed to L. A. Caffarelli.

6 recordsLinked to original sources
arXiv2024

A PDE approach to the existence and regularity of surfaces of minimum mean curvature variation

We develop an analytic theory of existence and regularity of surfaces (given by graphs) arising from the geometric minimization problem $$\min_{\mathcal{M}}\frac{1}{2}\int_{\mathcal{M}}|\nabla_{\mathcal{M}}H|^2\,dA$$ where $\mathcal{M}$ ranges over all $n$-dimensional manifolds in $\mathbb{R}^{n+1}$ with prescribed boundary, $\nabla_{\mathcal{M}}H$ is the tangential gradient along $\mathcal{M}$ of the mean curvature $H$ of $\mathcal{M}$ and $dA$ is the differential of surface area. The minimizers, called surfaces of minimum mean curvature variation, are central in applications of computer-aided design, computer-aided manufacturing and mechanics. Our main results show the existence of both smooth surfaces and of variational solutions to the minimization problem together with geometric regularity results. These are the first analytic results available on the literature for this problem.

math.DG↗
arXiv2022

Regularity for $C^{1,α}$ interface transmission problems

We study existence, uniqueness, and optimal regularity of solutions to transmission problems for harmonic functions with $C^{1,α}$ interfaces. For this, we develop a novel geometric stability argument based on the mean value property.

math.AP↗
arXiv2017

Fractional elliptic equations, Caccioppoli estimates and regularity

Let $L=-\operatorname{div}_x(A(x)\nabla_x)$ be a uniformly elliptic operator in divergence form in a bounded domain $Ω$. We consider the fractional nonlocal equations $$\begin{cases} L^su=f,&\hbox{in}~Ω,\\ u=0,&\hbox{on}~\partialΩ, \end{cases}\quad \hbox{and}\quad \begin{cases} L^su=f,&\hbox{in}~Ω,\\ \partial_Au=0,&\hbox{on}~\partialΩ. \end{cases}$$ Here $L^s$, $0<s<1$, is the fractional power of $L$ and $\partial_Au$ is the conormal derivative of $u$ with respect to the coefficients $A(x)$. We reproduce Caccioppoli type estimates that allow us to develop the regularity theory. Indeed, we prove interior and boundary Schauder regularity estimates depending on the smoothness of the coefficients $A(x)$, the right hand side $f$ and the boundary of the domain. Moreover, we establish estimates for fundamental solutions in the spirit of the classical result by Littman--Stampacchia--Weinberger and we obtain nonlocal integro-differential formulas for $L^su(x)$. Essential tools in the analysis are the semigroup language approach and the extension problem.

math.AP↗
arXiv2009

Nonlocal minimal surfaces

The de Giorgi theory for minimal surfaces consists in studying sets whose indicator function is a (local) minimum of the BV norm. In this paper we replace the BV norm by the $H^σ$ norm, with $σ<1/2$, and try to understand what the minimisers look like. Parallel to the de Giorgi theory we prove that, if the boundary of a minimiser is sufficiently flat in the unit ball, then it is a smooth piece of hypersurface.

math.AP↗