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Guido De Philippis

Publications and source records attributed to Guido De Philippis.

At least 37 records · Page 2Linked to original sources
arXiv2022

The anisotropic Min-Max theory: Existence of anisotropic minimal and CMC surfaces

We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth $3$-dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard [Invent. Math.,1983] in dimension $3$.

math.DG↗
arXiv2022

Closed geodesics on reversible Finsler 2-spheres

We extend two celebrated theorems on closed geodesics of Riemannian 2-spheres to the larger class of reversible Finsler 2-spheres: Lusternik-Schnirelmann's theorem asserting the existence of three simple closed geodesics, and Bangert-Franks-Hingston's theorem asserting the existence of infinitely many closed geodesics. In order to prove the first theorem, we employ the generalization of Grayson's curve shortening flow developed by Angenent-Oaks.

math.DG↗
arXiv2021

(Quasi-)conformal methods in two-dimensional free boundary problems

In this paper we study the local behavior of solutions to some free boundary problems. We relate the theory of quasi-conformal maps to the regularity of the solutions to nonlinear thin-obstacle problems; we prove that the contact set is locally a finite union of intervals and we apply this result to the solutions of one-phase Bernoulli problems with geometric constraint. We also introduce a new conformal hodograph transform, which allows to obtain the precise expansion at branch points of both the solutions to the one-phase problem with geometric constraint and a class of symmetric solutions to the two-phase problem, as well as to construct examples of free boundaries with cusp-like singularities.

math.AP↗
arXiv2021

Rectifiability and almost everywhere uniqueness of the blow-up for the vectorial Bernoulli free boundaries

We prove that for minimizers of the vectorial Alt-Caffarelli functional the two-phase singular set of the free boundary is rectifiable and the blow-up is unique almost everywhere on it. While the first conclusion is an application of the recent techniques developed by Naber and Valtorta, the uniqueness part follows from the rectifiability and a new application of the Alt-Caffarelli-Friedman monotonicity formula.

math.AP↗
arXiv2021

On the boundary behavior of mass-minimizing integral currents

Let $Σ$ be a smooth Riemannian manifold, $Γ\subset Σ$ a smooth closed oriented submanifold of codimension higher than $2$ and $T$ an integral area-minimizing current in $Σ$ which bounds $Γ$. We prove that the set of regular points of $T$ at the boundary is dense in $Γ$. Prior to our theorem the existence of any regular point was not known, except for some special choice of $Σ$ and $Γ$. As a corollary we answer to a question of Almgren about the connectivity of minimizers.

math.AP↗
arXiv2020

Fine properties of functions of bounded deformation -- an approach via linear PDEs

In this survey we collect some recent results obtained by the authors and collaborators concerning the fine structure of functions of bounded deformation (BD). These maps are $\mathrm{L}^1$-functions with the property that the symmetric part of their distributional derivative is representable as a bounded (matrix-valued) Radon measure. It has been known for a long time that for a (matrix-valued) Radon measure the property of being a symmetrized gradient can be characterized by an under-determined second-order PDE system, the Saint-Venant compatibility conditions. This observation gives rise to a new approach to the fine properties of BD-maps via the theory of PDEs for measures, which complements and partially replaces classical arguments. Starting from elementary observations, here we elucidate the ellipticity arguments underlying this recent progress and give an overview of the state of the art. We also present some open problems.

math.AP↗
arXiv2019

Positive solutions to the sublinear Lane-Emden equation are isolated

We prove that on a smooth bounded set, the positive least energy solution of the Lane-Emden equation with sublinear power is isolated. As a corollary, we obtain that the first $q-$eigenvalue of the Dirichlet-Laplacian is not an accumulation point of the $q-$spectrum, on a smooth bounded set. Our results extend to a suitable class of Lipschitz domains, as well.

math.AP↗
arXiv2019

Regularity of the free boundary for the two-phase Bernoulli problem

We prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As a consequence, we also show regularity of minimizers of the multiphase spectral optimization problem for the principal eigenvalue of the Dirichlet Laplacian.

math.AP↗
arXiv2019

Geometric measure theory and differential inclusions

In this paper we consider Lipschitz graphs of functions which are stationary points of strictly polyconvex energies. Such graphs can be thought as integral currents, resp. varifolds, which are stationary for some elliptic integrands. The regularity theory for the latter is a widely open problem, in particular no counterpart of the classical Allard's theorem is known. We address the issue from the point of view of differential inclusions and we show that the relevant ones do not contain the class of laminates which are used in [22] and [25] to construct nonregular solutions. Our result is thus an indication that an Allard's type result might be valid for general elliptic integrands. We conclude the paper by listing a series of open questions concerning the regularity of stationary points for elliptic integrands.

math.AP↗
arXiv2019

The behavior of harmonic functions at singular points of $\mathsf{RCD}$ spaces

In this note we investigate the behavior of harmonic functions at singular points of $\mathsf{RCD}(K,N)$ spaces. In particular we show that their gradient vanishes at all points where the tangent cone is isometric to a cone over a metric measure space with non-maximal diameter. The same analysis is performed for functions with a Laplacian in $L^{N+\varepsilon}$. As a consequence we show that on smooth manifolds there is no a priori estimate on the modulus of continuity of the gradient of harmonic functions which depends only on lower bounds of the sectional curvature. In the same way we show that there is no a priori Calderón-Zygmund theory for the Laplacian with bounds depending only on lower bounds of the sectional curvature.

math.DG↗
arXiv2019

Nonclassical minimizing surfaces with smooth boundary

We construct a Riemannian metric $g$ on $\mathbb{R}^4$ (arbitrarily close to the euclidean one) and a smooth simple closed curve $Γ\subset \mathbb R^4$ such that the unique area minimizing surface spanned by $Γ$ has infinite topology. Furthermore the metric is almost Kähler and the area minimizing surface is calibrated.

math.DG↗
arXiv2019

Implicit time discretization for the mean curvature flow of mean convex sets

In this note we analyze the Almgren-Taylor-Wang scheme for mean curvature flow in the case of mean convex initial conditions. We show that the scheme preserves strict mean convexity and, by compensated compactness techniques, that the arrival time functions converge strictly in \(BV\). In particular, this establishes the convergence of the time-integrated perimeters of the approximations. As a corollary, the conditional convergence result of Luckhaus-Sturzenhecker becomes unconditonal in the mean convex case.

math.AP↗
arXiv2019

Existence results for minimizers of parametric elliptic functionals

We prove a compactness principle for the anisotropic formulation of the Plateau problem in any codimension, in the same spirit of the previous works of the authors \cite{DelGhiMag,DePDeRGhi,DeLDeRGhi16}. In particular, we perform a new strategy for the proof of the rectifiability of the minimal set, based on the new anisotropic counterpart of the Allard rectifiability theorem proved by the authors in \cite{DePDeRGhi2}. As a consequence we provide a new proof of Reifenberg existence theorem.

math.AP↗
arXiv2019

The sharp quantitative isocapacitary inequality

We prove a sharp quantitative form of the classical isocapacitary inequality. Namely, we show that the difference between the capacity of a set and that of a ball with the same volume bounds the square of the Fraenkel asymmetry of the set. This provides a positive answer to a conjecture of Hall, Hayman, and Weitsman (J. d' Analyse Math. '91).

math.AP↗
arXiv2019

Dimensional estimates and rectifiability for measures satisfying linear PDE constraints

We establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators, including all first-order systems and all second-order scalar operators. In particular, our general theorem provides a new proof of the rectifiability results for functions of bounded variations (BV) and functions of bounded deformation (BD). For divergence-free tensors we obtain refinements and new proofs of several known results on the rectifiability of varifolds and defect measures.

math.AP↗
arXiv2019

The Area Blow Up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals

In this paper we investigate the "area blow-up" set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in (J. Differential Geom., 2016), we show that this set has bounded (anisotropic) mean curvature in the viscosity sense. In particular, this allows to show that the set is empty in a variety of situations. As a consequence, we show boundary curvature estimates for two dimensional stable anisotropic minimal surfaces, extending the results of (Invent. Math., 1987).

math.AP↗
arXiv2019

Regularity of minimizers for a model of charged droplets

We investigate properties of minimizers of a variational model describing the shape of charged liquid droplets. The model, proposed by Muratov and Novaga, takes into account the regularizing effect due to the screening of free counterionions in the droplet. In particular we prove partial regularity of minimizers, a first step toward the understanding of further properties of minimizers.

math.AP↗