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Guido De Philippis

Publications and source records attributed to Guido De Philippis.

At least 73 records · Page 4Linked to original sources
arXiv2015

A direct approach to Plateau's problem in any codimension

This paper aims to propose a direct approach to solve the Plateau's problem in codimension higher than one. The problem is formulated as the minimization of the Hausdorff measure among a family of $d$-rectifiable closed subsets of $\mathbb R^n$: following the previous work \cite{DelGhiMag} the existence result is obtained by a compactness principle valid under fairly general assumptions on the class of competitors. Such class is then specified to give meaning to boundary conditions. We also show that the obtained minimizers are regular up to a set of dimension less than $(d-1)$.

math.AP↗
arXiv2014

Existence of isoperimetric sets with densities "converging from below" in $\mathbb{R}^N$

In this paper, we consider the isoperimetric problem in the space $\mathbb{R}^N$ with density. Our result states that, if the density f is l.s.c. and converges to a positive limit at infinity, being smaller than this limit far from the origin, then isoperimetric sets exist for all volumes. Several known results or counterexamples show that the present result is essentially sharp. The special case of our result for radial and increasing densities positively answers a conjecture made in [10].

math.AP↗
arXiv2014

Regularity of free boundaries in anisotropic capillarity problems and the validity of Young's law

Local volume-constrained minimizers in anisotropic capillarity problems develop free boundaries on the walls of their containers. We prove the regularity of the free boundary outside a closed negligible set, showing in particular the validity of Young's law at almost every point of the free boundary. Our regularity results are not specific to capillarity problems, and actually apply to sets of finite perimeter (and thus to codimension one integer rectifiable currents) arising as minimizers in other variational problems with free boundaries.

math.AP↗
arXiv2013

Existence and regularity of minimizers for some spectral functionals with perimeter constraint

In this paper we prove that the shape optimization problem $$\min\left\{λ_k(Ω):\ Ω\subset\R^d,\ Ω \hbox{open},\ P(Ω)=1,\ |Ω|<+\infty\right\},$$ has a solution for any $k\in\N$ and dimension $d$. Moreover, every solution is a bounded connected open set with boundary which is $C^{1,α}$ outside a closed set of Hausdorff dimension $d-8$. Our results are more general and apply to spectral functionals of the form $f(λ_{k_1}(Ω),\dots,λ_{k_p}(Ω))$, for increasing functions $f$ satisfying some suitable bi-Lipschitz type condition.

math.AP↗
arXiv2013

Rigidity of equality cases in Steiner's perimeter inequality

Characterization results for equality cases and for rigidity of equality cases in Steiner's perimeter inequality are presented. (By rigidity, we mean the situation when all equality cases are vertical translations of the Steiner's symmetral under consideration.) We achieve this through the introduction of a suitable measure-theoretic notion of connectedness and a fine analysis of barycenter functions for sets of finite perimeter having segments as orthogonal sections with respect to an hyperplane.

math.AP↗
arXiv2013

Faber-Krahn inequalities in sharp quantitative form

The classical Faber-Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet-Laplacian among sets with given volume. In this paper we prove a sharp quantitative enhancement of this result, thus confirming a conjecture by Nadirashvili and Bhattacharya-Weitsman. More generally, the result applies to every optimal Poincaré-Sobolev constant for the embeddings $W^{1,2}_0(Ω)\hookrightarrow L^q(Ω)$.

math.AP↗
arXiv2012

Sobolev regularity for Monge-Ampère type equations

In this note we prove that, if the cost function satisfies some necessary structural conditions and the densities are bounded away from zero and infinity, then strictly $c$-convex potentials arising in optimal transportation belong to $W^{2,1+κ}_{\rm loc}$ for some $κ>0$. This generalizes some recents results concerning the regularity of strictly convex Alexandrov solutions of the Monge-Ampère equation with right hand side bounded away from zero and infinity.

math.AP↗
arXiv2012

Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case

In this paper we use the new regularity and stability estimates for Alexandrov solutions to Monge-Ampere equations estabilished by G.De Philippis and A.Figalli to provide a global in time existence of distributional solutions to a semigeostrophic equation on the 2-dimensional torus, under very mild assumptions on the initial data. A link with Lagrangian solutions is also discussed.

math.AP↗
arXiv2012

$W^{2,1}$ regularity for solutions of the Monge-Ampère equation

In this paper we prove that a strictly convex Alexandrov solution u of the Monge-Ampère equation, with right hand side bounded away from zero and infinity, is $W_{\rm loc}^{2,1}$. This is obtained by showing higher integrability a-priori estimates for $D^2 u$, namely $D^2 u \in L\log^k L$ for any $k\in \mathbb N$.

math.AP↗
arXiv2011

On the Aleksandrov-Bakelman-Pucci estimate for the infinity Laplacian

We prove $L^\infty$ bounds and estimates of the modulus of continuity of solutions to the Poisson problem for the normalized infinity and $p$-Laplacian, namely \[ -Δ_p^N u=f\qquad\text{for $n<p\leq\infty$.} \] We are able to provide a stable family of results depending continuously on the parameter $p$. We also prove the failure of the classical Alexandrov-Bakelman-Pucci estimate for the normalized infinity Laplacian and propose alternate estimates.

math.AP↗