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Giorgio Poggesi

Publications and source records attributed to Giorgio Poggesi.

31 records · Page 2Linked to original sources
arXiv2021

Interpolating estimates with applications to some quantitative symmetry results

We prove interpolating estimates providing a bound for the oscillation of a function in terms of two $L^p$ norms of its gradient. They are based on a pointwise bound of a function on cones in terms of the Riesz potential of its gradient. The estimates hold for a general class of domains, including, e.g., Lipschitz domains. All the constants involved can be explicitly computed. As an application, we show how to use these estimates to obtain stability for Alexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary value problem. The new approach results in several novelties and benefits for these problems.

math.AP↗
arXiv2021

An interpolating inequality for solutions of uniformly elliptic equations

We extend an inequality for harmonic functions, obtained in previous research by the authors, to the case of solutions of uniformly elliptic equations in divergence form, with merely measurable coefficients. The inequality for harmonic functions turned out to be a crucial ingredient in the study of the stability of the radial symmetry for Alexandrov's Soap Bubble Theorem and Serrin's problem. The proof of our inequality is based on a mean value property for elliptic operators by L. Caffarelli, proved in full detail by I. Blank and Z. Hao.

math.AP↗
arXiv2021

Quantitative stability estimates for a two-phase Serrin-type overdetermined problem

In this paper, we deal with an overdetermined problem of Serrin-type with respect to a two-phase elliptic operator in divergence form with piecewise constant coefficients. In particular, we consider the case where the two-phase overdetermined problem is close to the one-phase setting. First, we show quantitative stability estimates for the two-phase problem via a one-phase stability result. Furthermore, we prove non-existence for the corresponding inner problem by the aforementioned two-phase stability result.

math.AP↗
arXiv2021

A quantitative rigidity result for a two-dimensional Frenkel-Kontorova model

We consider a Frenkel-Kontorova system of harmonic oscillators in a two-dimensional Euclidean lattice and we obtain a quantitative estimate on the angular function of the equilibria. The proof relies on a PDE method related to a classical conjecture by E. De Giorgi, also in view of an elegant technique based on complex variables that was introduced by A. Farina. In the discrete setting, a careful analysis of the reminders is needed to exploit this type of methodologies inspired by continuum models.

math.AP↗
arXiv2020

The location of hot spots and other extremal points

In a domain of the Euclidean space, we estimate from below the distance to the boundary of global maximum points of solutions of elliptic and parabolic equations with homogeneous Dirichlet boundary values. As reference cases, we first consider the torsional rigidity function of a bar, the first mode of a vibrating membrane, and the temperature of a heat conductor grounded to zero at the boundary. Our main results are presented for domains with a mean convex boundary and compare that distance to the inradius of the relevant domain. For the torsional rigidity function, the obtained bound only depends on the space dimension. The more general case of a boundary which is not mean convex is also considered. However, the estimates also depend on some geometrical quantities such as the diameter and the radius of the largest exterior osculating ball to the relevant domain, or the minimum of the mean curvature of the boundary. Also in the case of the first mode, the relevant bound only depends on the space dimension. Moreover, it largely improves on an earlier estimate obtained by the first author and co-authors, for convex domains. The bound related to the temperature depends on time and the initial distribution of temperature. Such a bound is substantially consistent with what one obtains in the stationary situation. The methods employed are based on elementary arguments and existing literature, and can be extended to other situations that entail quasilinear equations, isotropic and anisotropic, and also certain classes of semilinear equations.

math.AP↗
arXiv2020

A Serrin-type problem with partial knowledge of the domain

We present a quantitative estimate for the radially symmetric configuration concerning a Serrin-type overdetermined problem for the torsional rigidity in a bounded domain $Ω$, when the equation is known on $Ω\setminus \overlineω$ only, for some open subset $ω\Subset Ω$. The problem has concrete motivations in optimal heating with malfunctioning, laminar flows and beams with small inhomogeneities.

math.AP↗
arXiv2019

Nearly optimal stability for Serrin's problem and the Soap Bubble theorem

We present new quantitative estimates for the radially symmetric configuration concerning Serrin's overdetermined problem for the torsional rigidity, Alexandrov's Soap Bubble Theorem, and other related problems. The new estimates improve on those obtained in two our previous articles, and are in some cases optimal.

math.AP↗
arXiv2018

Stable solutions to some elliptic problems: minimal cones, the Allen-Cahn equation, and blow-up solutions

These notes record the lectures for the CIME Summer Course taught by the first author in Cetraro during the week of June 19-23, 2017. The notes contain the proofs of several results on the classification of stable solutions to some nonlinear elliptic equations. The results are crucial steps within the regularity theory of minimizers to such problems. We focus our attention on three different equations, emphasizing that the techniques and ideas in the three settings are quite similar. The first topic is the stability of minimal cones. We prove the minimality of the Simons cone in high dimensions, and we give almost all details in the proof of J. Simons on the flatness of stable minimal cones in low dimensions. Its semilinear analogue is a conjecture on the Allen-Cahn equation posed by E. De Giorgi in 1978. This is our second problem, for which we discuss some results, as well as an open problem in high dimensions on the saddle-shaped solution vanishing on the Simons cone. The third problem was raised by H. Brezis around 1996 and concerns the boundedness of stable solutions to reaction-diffusion equations in bounded domains. We present proofs on their regularity in low dimensions and discuss the main open problem in this topic. Moreover, we briefly comment on related results for harmonic maps, free boundary problems, and nonlocal minimal surfaces.

math.AP↗
arXiv2018

Radial symmetry for p-harmonic functions in exterior and punctured domains

We prove symmetry for the p-capacitary potential satisfying $$ Δ_p u = 0 \, \mbox{ in } \mathbb{R}^N \setminus \overlineΩ , \; u=1 \, \mbox{ on } Γ, \; \lim_{|x|\rightarrow \infty} u(x)=0 , \; \; \; \; \; \; \; \; 1<p<N, $$ under Serrin's overdetermined condition $$ | \nabla u| = c \mbox{ on } Γ. $$ Here $Ω$ is any bounded domain on which no a priori assumption is made, and $Γ$ denotes its boundary. Our result improves on a work of Garofalo and Sartori, where the same conclusion was obtained when $Ω$ is star-shaped. Our proof uses the maximum principle for an appropriate $P$-function, some integral identities, the isoperimetric inequality, and a Soap Bubble-type Theorem. We then treat the case $1<p=N$, improving previous results present in the literature. Finally, with analogous tools we give a new proof of symmetry for the interior overdetermined problem $$ - Δ_p u = K \, δ_0 \, \mbox{ in } Ω, \, u=c \, \mbox{ on } Γ, \; \; \; \; \; \; \; \; 1<p<N, $$ $$ | \nabla u| = 1 \mbox{ on } Γ, $$ in a bounded star-shaped domain $Ω$.

math.AP↗
arXiv2017

Serrin's problem and Alexandrov's Soap Bubble Theorem: enhanced stability via integral identities

We consider Serrin's overdetermined problem for the torsional rigidity and Alexandrov's Soap Bubble Theorem. We present new integral identities, that show a strong analogy between the two problems and help to obtain better (in some cases optimal) quantitative estimates for the radially symmetric configuration. The estimates for the Soap Bubble Theorem benefit from those of Serrin's problem.

math.AP↗
arXiv2017

On the stability for Alexandrov's Soap Bubble theorem

Alexandrov's Soap Bubble theorem dates back to $1958$ and states that a compact embedded hypersurface in $\mathbb{R}^N$ with constant mean curvature must be a sphere. For its proof, A.D. Alexandrov invented his reflection priciple. In $1982$, R. Reilly gave an alternative proof, based on integral identities and inequalities, connected with the torsional rigidity of a bar. In this article we study the stability of the spherical symmetry: the question is how much a hypersurface is near to a sphere, when its mean curvature is near to a constant in some norm. We present a stability estimate that states that a compact hypersurface $Γ\subset\mathbb{R}^N$ can be contained in a spherical annulus whose interior and exterior radii, say $ρ_i$ and $ρ_e$, satisfy the inequality $$ ρ_e - ρ_i \le C \Vert H - H_0 \Vert^{τ_N}_{L^1 (Γ)}, $$ where $τ_N=1/2$ if $N=2, 3$, and $τ_N=1/(N+2)$ if $N\ge 4$. Here, $H$ is the mean curvature of $Γ$, $H_0$ is some reference constant and $C$ is a constant that depends on some geometrical and spectral parameters associated with $Γ$. This estimate improves previous results in the literature under various aspects. We also present similar estimates for some related overdetermined problems.

math.AP↗
arXiv2014

Littlewood's fourth principle

In Real Analysis, Littlewood's three principles are known as heuristics that help teach the essentials of measure theory and reveal the analogies between the concepts of topological space and continuos function on one side and those of measurable space and measurable function on the other one. They are based on important and rigorous statements, such as Lusin's and Egoroff-Severini's theorems, and have ingenious and elegant proofs. We shall comment on those theorems and show how their proofs can possibly be made simpler by introducing a \textit{fourth principle}. These alternative proofs make even more manifest those analogies and show that Egoroff-Severini's theorem can be considered the natural generalization of the classical Dini's monotone convergence theorem.

math.CA↗