arXiv subjects

Rolando Magnanini

Publications and source records attributed to Rolando Magnanini.

At least 37 records · Page 2Linked to original sources

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↗

arXiv2016

An introduction to the study of critical points of solutions of elliptic and parabolic equations

We give a survey at an introductory level of old and recent results in the study of critical points of solutions of elliptic and parabolic partial differential equations. To keep the presentation simple, we mainly consider four exemplary boundary value problems: the Dirichlet problem for the Laplace's equation; the torsional creep problem; the case of Dirichlet eigenfunctions for the Laplace's equation; the initial-boundary value problem for the heat equation. We shall mostly address three issues: the estimation of the local size of the critical set; the dependence of the number of critical points on the boundary values and the geometry of the domain; the location of critical points in the domain.

math.AP↗

arXiv2016

A natural approach to the asymptotic mean value property for the $p$-Laplacian

Let $1\le p\le\infty$. We show that a function $u\in C(\mathbb R^N)$ is a viscosity solution to the normalized $p$-Laplace equation $Δ_p^n u(x)=0$ if and only if the asymptotic formula $$ u(x)=μ_p(\ve,u)(x)+o(\ve^2) $$ holds as $\ve\to 0$ in the viscosity sense. Here, $μ_p(\ve,u)(x)$ is the $p$-mean value of $u$ on $B_\ve(x)$ characterized as a unique minimizer of $$ \inf_{\la\in\RR}\nr u-\la\nr_{L^p(B_\ve(x))}. $$ This kind of asymptotic mean value property (AMVP) extends to the case $p=1$ previous (AMVP)'s obtained when $μ_p(\ve,u)(x)$ is replaced by other kinds of mean values. The natural definition of $μ_p(\ve,u)(x)$ makes sure that this is a monotonic and continuous (in the appropriate topology) functional of $u$. These two properties help to establish a fairly general proof of (AMVP), that can also be extended to the (normalized) parabolic $p$-Laplace equation.

math.AP↗

arXiv2015

Hölder stability for Serrin's overdetermined problem

In a bounded domain $Ω$, we consider a positive solution of the problem $Δu+f(u)=0$ in $Ω$, $u=0$ on $\partialΩ$, where $f:\mathbb{R}\to\mathbb{R}$ is a locally Lipschitz continuous function. Under sufficient conditions on $Ω$ (for instance, if $Ω$ is convex), we show that $\partialΩ$ is contained in a spherical annulus of radii $r_i 0$ and $α\in (0,1]$. Here, $[u_ν]_{\partialΩ}$ is the Lipschitz seminorm on $\partialΩ$ of the normal derivative of $u$. This result improves to Hölder stability the logarithmic estimate obtained in [1] for Serrin's overdetermined problem. It also extends to a large class of semilinear equations the Hölder estimate obtained in [6] for the case of torsional rigidity ($f\equiv 1$) by means of integral identities. The proof hinges on ideas contained in [1] and uses Carleson-type estimates and improved Harnack inequalities in cones.

math.AP↗

arXiv2015

Symmetry and linear stability in Serrin's overdetermined problem via the stability of the parallel surface problem

We consider the solution of the problem $$ -Δu=f(u) \ \mbox{ and } \ u>0 \ \ \mbox{ in } \ Ω, \ \ u=0 \ \mbox{ on } \ Γ, $$ where $Ω$ is a bounded domain in $\mathbb{R}^N$ with boundary $Γ$ of class $C^{2,τ}$, $0<τ<1$, and $f$ is a locally Lipschitz continuous non-linearity. Serrin's celebrated symmetry theorem states that, if the normal derivative $u_ν$ is constant on $Γ$, then $Ω$ must be a ball. In [CMS2], it has been conjectured that Serrin's theorem may be obtained by stability in the following way: first, for a solution $u$ prove the estimate $$ r_e-r_i\le C_δ\,[u]_{Γ^δ} $$ for some constant $C_δ$ depending on $δ>0$, where $r_e$ and $r_i$ are the radii of a spherical annulus containing $Γ$, $Γ^δ$ is a surface parallel to $Γ$ at distance $δ$ and sufficiently close to $Γ$, and $[u]_{Γ^δ}$ is the Lipschitz semi-norm of $u$ on $Γ^δ$; secondly, if in addition $u_ν$ is constant on $Γ$, show that $$ [u]_{Γ^δ}=o(C_δ)\ \mbox{ as } \ δ\to 0^+. $$ In this paper, we prove that this strategy is successful. As a by-product of this method, for $C^{2,τ}$-regular domains, we also obtain a linear stability estimate for Serrin's symmetry result. Our result is optimal and greatly improves the similar logarithmic-type estimate of [ABR] and the Hölder estimate of [CMV] that was restricted to convex domains.

math.AP↗

arXiv2015

The Matzoh Ball Soup Problem: a complete characterization

We characterize all the solutions of the heat equation that have their (spatial) equipotential surfaces which do not vary with the time. Such solutions are either isoparametric or split in space-time. The result gives a final answer to a problem raised by M. S. Klamkin, extended by G. Alessandrini, and that was named the Matzoh Ball Soup Problem by L. Zalcman. Similar results can also be drawn for a class of quasi-linear parabolic partial differential equations with coefficients which are homogeneous functions of the gradient variable. This class contains the (isotropic or anisotropic) evolution p-Laplace and normalized p-Laplace equations.

math.AP↗

arXiv2015

Stationary isothermic surfaces in Euclidean 3-space

Let $Ω$ be a domain in $\mathbb R^3$ with $\partialΩ= \partial\left(\mathbb R^3\setminus \overlineΩ\right)$, where $\partialΩ$ is unbounded and connected, and let $u$ be the solution of the Cauchy problem for the heat equation $\partial_t u= Δu$ over $\mathbb R^3,$ where the initial data is the characteristic function of the set $Ω^c = \mathbb R^3\setminus Ω$. We show that, if there exists a stationary isothermic surface $Γ$ of $u$ with $Γ\cap \partialΩ= \varnothing$, then both $\partialΩ$ and $Γ$ must be either parallel planes or co-axial circular cylinders . This theorem completes the classification of stationary isothermic surfaces in the case that $Γ\cap\partialΩ=\varnothing$ and $\partialΩ$ is unbounded. To prove this result, we establish a similar theorem for {\it uniformly dense domains } in $\mathbb R^3$, a notion that was introduced by Magnanini, Prajapat \& Sakaguchi in \cite{MPS2006tams}. In the proof, we use methods from the theory of surfaces with constant mean curvature, combined with a careful analysis of certain asymptotic expansions and a surprising connection with the theory of transnormal functions.

math.AP↗

arXiv2014

Symmetry of minimizers with a level surface parallel to the boundary

We consider the functional $$I_Ω(v) = \int_Ω[f(|Dv|) - v] dx,$$ where $Ω$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$, G. Crasta [Cr1] has shown that if $I_Ω$ admits a minimizer in $W_0^{1,1}(Ω)$ depending only on the distance from the boundary of $Ω$, then $Ω$ must be a ball. With some restrictions on $f$, we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these results extend to more general settings, in particular to functionals that are not differentiable and to solutions of fully nonlinear elliptic and parabolic equations.

math.AP↗

arXiv2014

The heart of a convex body

We investigate some basic properties of the {\it heart} $\heartsuit(\mathcal{K})$ of a convex set $\mathcal{K}.$ It is a subset of $\mathcal{K},$ whose definition is based on mirror reflections of euclidean space, and is a non-local object. The main motivation of our interest for $\heartsuit(\mathcal{K})$ is that this gives an estimate of the location of the hot spot in a convex heat conductor with boundary temperature grounded at zero. Here, we investigate on the relation between $\heartsuit(\mathcal{K})$ and the mirror symmetries of $\mathcal{K};$ we show that $\heartsuit(\mathcal{K})$ contains many (geometrically and phisically) relevant points of $\mathcal{K};$ we prove a simple geometrical lower estimate for the diameter of $\heartsuit(\mathcal{K});$ we also prove an upper estimate for the area of $\heartsuit(\mathcal{K}),$ when $\mathcal{K}$ is a triangle.

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↗

arXiv2014

A note on Serrin's overdetermined problem

We consider the solution of the torsion problem $-Δu=1$ in $Ω$ and $u=0$ on $\partial Ω$. Serrin's celebrated symmetry theorem states that, if the normal derivative $u_ν$ is constant on $\partial Ω$, then $Ω$ must be a ball. In a recent paper, it has been conjectured that Serrin's theorem may be obtained {\it by stability} in the following way: first, for the solution $u$ of the torsion problem prove the estimate $$ r_e-r_i\leq C_t\,\Bigl(\max_{Γ_t} u-\min_{Γ_t} u\Bigr) $$ for some constant $C_t$ depending on $t$, where $r_e$ and $r_i$ are the radii of an annulus containing $\partialΩ$ and $Γ_t$ is a surface parallel to $\partialΩ$ at distance $t$ and sufficiently close to $\partialΩ$; secondly, if in addition $u_ν$ is constant on $\partialΩ$, show that $$ \max_{Γ_t} u-\min_{Γ_t} u=o(C_t)\ \mbox{as} \ t\to 0^+. $$ In this paper, we analyse a simple case study and show that the scheme is successful if the admissible domains $Ω$ are ellipses.

math.AP↗

arXiv2013

Characterization of ellipsoids as K-dense sets

Let K\subset R^N be any convex body containing the origin. A measurable set G\subset R^N with finite and positive Lebesgue measure is said to be K-dense if, for any fixed r>0, the measure of G\cap (x+r K) is constant when x varies on the boundary of G (here, x+r K denotes a translation of a dilation of K). In [6], we proved for the case in which N=2 that if G is K-dense, then both G and K must be homothetic to the same ellipse. Here, we completely characterize K-dense sets in R^N: if G is K-dense, then both G and K must be homothetic to the same ellipsoid. Our proof, by building upon results obtained in [6], relies on an asymptotic formula for the measure of G\cap (x+r K) for large values of the parameter r and a classical characterization of ellipsoids due to C.M. Petty [9].

math.MG↗

arXiv2013

Characterization of ellipses as uniformly dense sets with respect to a family of convex bodies

Let K \subset R^N be a convex body containing the origin. A measurable set G \subset R^N with positive Lebesgue measure is said to be uniformly K-dense if, for any fixed r > 0, the measure of G \cap (x + rK) is constant when x varies on the boundary of G (here, x + rK denotes a translation of a dilation of K). We first prove that G must always be strictly convex and at least C1,1-regular; also, if K is centrally symmetric, K must be strictly convex, C1,1-regular and such that K = G - G up to homotheties; this implies in turn that G must be C2,1- regular. Then for N = 2, we prove that G is uniformly K-dense if and only if K and G are homothetic to the same ellipse. This result was already proven by Amar, Berrone and Gianni in [3]. However, our proof removes their regularity assumptions on K and G and, more importantly, it is susceptible to be generalized to higher dimension since, by the use of Minkowski's inequality and an affine inequality, avoids the delicate computations of the higher-order terms in the Taylor expansion near r = 0 for the measure of G\cap(x+rK) (needed in [3]).

math.MG↗

arXiv2013

Solutions of elliptic equations with a level surface parallel to the boundary: stability of the radial configuration

Positive solutions of homogeneous Dirichlet boundary value problems or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of their level surfaces is parallel to the boundary of the domain. Here, for the elliptic case, we prove the stability counterpart of that result. In fact, we show that if the solution is almost constant on a surface at a fixed distance from the boundary, then the domain is almost radially symmetric, in the sense that is contained in and contains two concentric balls $B_{r_e}$ and $B_{r_i}$, with the difference $r_e-r_i$ (linearly) controlled by a suitable norm of the deviation of the solution from a constant. The proof relies on and enhances arguments developed in a paper by Aftalion, Busca and Reichel.

math.AP↗

arXiv2011

Interaction between nonlinear diffusion and geometry of domain

Let $Ω$ be a domain in $\mathbb R^N$, where $N \ge 2$ and $\partialΩ$ is not necessarily bounded. We consider nonlinear diffusion equations of the form $\partial_t u= Δϕ(u)$. Let $u=u(x,t)$ be the solution of either the initial-boundary value problem over $Ω$, where the initial value equals zero and the boundary value equals 1, or the Cauchy problem where the initial data is the characteristic function of the set $\mathbb R^N\setminus Ω$. We consider an open ball $B$ in $Ω$ whose closure intersects $\partialΩ$ only at one point, and we derive asymptotic estimates for the content of substance in $B$ for short times in terms of geometry of $Ω$. Also, we obtain a characterization of the hyperplane involving a stationary level surface of $u$ by using the sliding method due to Berestycki, Caffarelli, and Nirenberg. These results tell us about interactions between nonlinear diffusion and geometry of domain.

math.AP↗

arXiv2011

Matzoh ball soup revisited: the boundary regularity issue

We consider nonlinear diffusion equations of the form $\partial_t u= Δϕ(u)$ in $\mathbb R^N$ with $N \ge 2.$ When $ϕ(s) \equiv s$, this is just the heat equation. Let $Ω$ be a domain in $\mathbb R^N$, where $\partialΩ$ is bounded and $\partialΩ= \partial (\mathbb R^N\setminus \bar Ω)$. We consider the initial-boundary value problem, where the initial value equals zero and the boundary value equals 1, and the Cauchy problem where the initial data is the characteristic function of the set $Ω^c = \mathbb R^N\setminus Ω$. We settle the boundary regularity issue for the characterization of the sphere as a stationary level surface of the solution $u:$ no regularity assumption is needed for $\partialΩ.$

math.AP↗

arXiv2010

The location of the hot spot in a grounded convex conductor

We investigate the location of the (unique) hot spot in a convex heat conductor with unitary initial temperature and with boundary grounded at zero temperature. We present two methods to locate the hot spot: the former is based on ideas related to the Alexandrov-Bakelmann-Pucci maximum principle and Monge-Ampère equations; the latter relies on Alexandrov's reflection principle. We then show how such a problem can be simplified in case the conductor is a polyhedron. Finally, we present some numerical computations.

math.AP↗