arXiv subjects

Giorgio Poggesi

Publications and source records attributed to Giorgio Poggesi.

At least 19 recordsLinked to original sources

arXiv2025

Optimal quantitative stability estimates for Alexandrov's Soap Bubble Theorem via Gagliardo-Nirenberg-type interpolation inequalities

The paper provides optimal quantitative stability estimates for the celebrated Alexandrov's Soap Bubble Theorem within the class of $C^{k,α}$ domains, for any $k \ge 1$ and $0 < α\leq 1$, by leveraging Gagliardo-Nirenberg-type interpolation inequalities. Optimal estimates of uniform closeness to a ball are established for $L^r$ deviations of the mean curvature from being constant, for any $r\geq 2$ (more generally, for any $r>1$ such that $r\geq (2N-2)/(N+1)$). For $r>\frac{N-1}{2}$, the stability profile is linear, thus returning the existing results established in the literature through computations for nearly spherical sets. All the stability estimates for $r\le \frac{N-1}{2}$, for which the profile is not linear, are new; even in the particular case $r=2$ (which has been extensively studied, since it is a case of interest for several critical applications), the sharp stability profile that we obtain is new. Interestingly, we also prove that the (non-linear) profile for $r \leq \frac{N-1}{2}$ improves as $k$ becomes larger to such an extent that it becomes formally linear as $k$ goes to $\infty$. Finally, for any $k \geq 1$ and $0< α\leq 1$, we show that our estimates are optimal within the class of $C^{k,α}$ domains, by providing explicit examples.

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arXiv2025

Bubbling and quantitative stability for Alexandrov's Soap Bubble Theorem with $L^1$-type deviations

The quantitative analysis of bubbling phenomena for almost constant mean curvature boundaries is an important question having significant applications in various fields including capillarity theory and the study of mean curvature flows. Such a quantitative analysis was initiated in [G. Ciraolo and F. Maggi, Comm. Pure Appl. Math. (2017)], where the first quantitative result of proximity to a set of disjoint balls of equal radii was obtained in terms of a uniform deviation of the mean curvature from being constant. Weakening the measure of the deviation in such a result is a delicate issue that is crucial in view of the applications for mean curvature flows. Some progress in this direction was recently made in [V. Julin and J. Niinikoski, Anal. PDE (2023)], where $L^{N-1}$-deviations are considered for domains in $\mathbb{R}^N$. In the present paper we significantly weaken the measure of the deviation, obtaining a quantitative result of proximity to a set of disjoint balls of equal radii for the following deviation $$ \int_{\partial Ω} \left( H_0 - H \right)^+ dS_x, \quad \text{ where } \begin{cases} H \text{ is the mean curvature of } \partial Ω, \\ H_0:=\frac{| \partial Ω|}{N | Ω|} , \\ \left( H_0 - H \right)^+:=\max\left\lbrace H_0 - H , 0 \right\rbrace , \end{cases} $$ which is clearly even weaker than $\Vert H_0-H \Vert_{L^1( \partial Ω)}$.

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arXiv2025

Hessian operators, overdetermined problems, and higher order mean curvatures: symmetry and stability results

It is well known that there is a deep connection between Serrin's symmetry result -- dealing with overdetermined problems involving the Laplacian -- and the celebrated Alexandrov's Soap Bubble Theorem (SBT) -- stating that, if the mean curvature $H$ of the boundary of a smooth bounded connected open set $\Om$ is constant, then $\Om$ must be a ball. One of the main aims of the paper is to extend the study of such a connection to the broader case of overdetermined problems for Hessian operators and constant higher order mean curvature boundaries. Our analysis will not only provide new proofs of the higher order SBT (originally established by Alexandrov) and of the symmetry for overdetermined Serrin-type problems for Hessian equations (originally established by Brandolini, Nitsch, Salani, and Trombetti), but also bring several benefits, including new interesting symmetry results and quantitative stability estimates. In fact, leveraging the analysis performed in the classical case (i.e., with classical mean curvature and classical Laplacian) by Magnanini and Poggesi in a series of papers, we will extend their approach to the higher order setting (i.e., with $k$-order mean curvature and $k$-Hessian operator, for $k \ge 1$) achieving various quantitative estimates of closeness to the symmetric configuration. Finally, leveraging the quantitative analysis in presence of bubbling phenomena performed in arXiv:2405.06376, we also provide a quantitative stability result of closeness of almost constant $k$-mean curvature boundaries to a set given by the union of a finite number of disjoint balls of equal radii. In passing, we will also provide two alternative proofs of the result established by Brandolini, Nitsch, Salani, and Trombetti, one of which provides the extension to Hessian operators of the approach famously pioneered by Weinberger for the classical Laplacian.

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arXiv2025

Elliptic and parabolic overdetermined problems in multi-phase settings

The present paper provides symmetry results for a class of overdetermined problems of elliptic and parabolic type in multi-phase settings, including various extensions of remarkable results obtained by S. Sakaguchi in [12, 13]. A new alternative approach to proving this type of results is presented, leveraging the weak formulation of the problem. The resulting proofs are direct and elegant, and bring several benefits, including extensions to multi-phase settings (possibly with infinitely many phases) and generalizations to rough interfaces.

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arXiv2025

Face 2-phase: how much overdetermination is enough to get symmetry in two-phase problems

We provide a full characterization of multi-phase problems under a large class of overdetermined Serrin-type conditions. Our analysis includes both symmetry and asymmetry (including bifurcation) results. A broad range of techniques is needed to obtain a full characterization of all the cases, including applications of results obtained via the moving planes method, approaches via integral identities in the wake of Weinberger, applications of the Crandall-Rabinowitz theorem, and the Chauchy-Kovalevskaya theorem. The multi-phase setting entails intrinsic difficulties that make it difficult to predict whether a given overdetermination will lead to symmetry or asymmetry results; the results of our analysis are significant as they answer such a question providing a full characterization of both symmetry and asymmetry results.

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arXiv2025

Fractional De Giorgi conjecture in dimension 2 via complex-plane methods

We provide a new proof of the fractional version of the De Giorgi conjecture for the Allen-Cahn equation in $\mathbb{R}^2$ for the full range of exponents. Our proof combines a method introduced by A. Farina in 2003 with the $s$-harmonic extension of the fractional Laplacian in the half-space $\mathbb{R}^{3}_+$ introduced by L. Caffarelli and L. Silvestre in 2007. We also provide a representation formula for finite-energy weak solutions of a class of weighted elliptic partial differential equations in the half-space $\mathbb{R}^{n+1}_+$ under Neumann boundary conditions. This generalizes the $s$-harmonic extension of the fractional Laplacian and allows us to relate a general problem in the extended space with a nonlocal problem on the trace.

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arXiv2024

A quantitative Gidas-Ni-Nirenberg-type result for the $p$-Laplacian via integral identities

We prove a quantitative version of a Gidas-Ni-Nirenberg-type symmetry result involving the $p$-Laplacian. Quantitative stability is achieved here via integral identities based on the proof of rigidity established by J. Serra in 2013, which extended to general dimension and the $p$-Laplacian operator an argument proposed by P. L. Lions in dimension $2$ for the classical Laplacian. Stability results for the classical Gidas-Ni-Nirenberg symmetry theorem (involving the classical Laplacian) via the method of moving planes were established by Rosset in 1994 and by Ciraolo, Cozzi, Perugini, Pollastro in 2024. To the authors' knowledge, the present paper provides the first quantitative Gidas-Ni-Nirenberg-type result involving the $p$-Laplacian for $p \neq 2$. Even for the classical Laplacian (i.e., for $p=2$), this is the first time that integral identities are used to achieve stability for a Gidas-Ni-Nirenberg-type result. In passing, we obtain a quantitative estimate for the measure of the singular set and an explicit uniform gradient bound.

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arXiv2024

A general integral identity with applications to a reverse Serrin problem

We prove a new general differential identity and an associated integral identity, which entails a pair of solutions of the Poisson equation with constant source term. This generalizes a formula that the first and third authors previously proved and used to obtain quantitative estimates of spherical symmetry for the Serrin overdetermined boundary value problem. As an application, we prove a quantitative symmetry result for the reverse Serrin problem, which we introduce for the first time in this paper. In passing, we obtain a rigidity result for solutions of the aforementioned Poisson equation subject to a constant Neumann condition.

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arXiv2023

Quantitative symmetry in a mixed Serrin-type problem for a constrained torsional rigidity

We consider a mixed boundary value problem in a domain $Ω$ contained in a half-ball $B_+$ and having a portion $\bar{T}$ of its boundary in common with the curved part of $\partial B_+$. The problem has to do with some sort of constrained torsional rigidity. In this situation, the relevant solution $u$ satisfies a Steklov condition on $T$ and a homogeneous Dirichlet condition on $Σ= \partialΩ\setminus \bar{T} \subset B_+$. We provide an integral identity that relates (a symmetric function of) the second derivatives of the solution in $Ω$ to its normal derivative $u_ν$ on $Σ$. A first significant consequence of this identity is a rigidity result under a quite weak overdetermining integral condition for $u_ν$ on $Σ$: in fact, it turns out that $Σ$ must be a spherical cap that meets $T$ orthogonally. This result returns the one obtained by J. Guo and C. Xia under the stronger pointwise condition that the values of $u_ν$ be constant on $Σ$. A second important consequence is a set of stability bounds, which quantitatively measure how $Σ$ is far uniformly from being a spherical cap, if $u_ν$ deviates from a constant in the norm $L^1(Σ)$.

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arXiv2023

Quantitative stability for the nonlocal overdetermined Serrin problem

We establish quantitative stability for the nonlocal Serrin overdetermined problem, via the method of the moving planes. Interestingly, our stability estimate is even better than those obtained so far in the classical setting (i.e., for the classical Laplacian) via the method of the moving planes. A crucial ingredient is the construction of a new antisymmetric barrier, which allows a unified treatment of the moving planes method. This strategy allows us to establish a new general quantitative nonlocal maximum principle for antisymmetric functions, leading to new quantitative nonlocal versions of both the Hopf lemma and the Serrin corner point lemma. All these tools -- i.e., the new antisymmetric barrier, the general quantitative nonlocal maximum principle, and the quantitative nonlocal versions of both the Hopf lemma and the Serrin corner point lemma -- are of independent interest.

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arXiv2023

Optimal quantitative stability for a Serrin-type problem in convex cones

We consider a Serrin-type problem in convex cones in the Euclidean space and motivated by recent rigidity results we study the quantitative stability issue for this problem. In particular, we prove both sharp Lipschitz estimates for an $L^2-$pseudodistance and estimates in terms of the Hausdorff distance.

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arXiv2023

Quantitative stability for overdetermined nonlocal problems with parallel surfaces and investigation of the stability exponents

In this article, we analyze the stability of the parallel surface problem for semilinear equations driven by the fractional Laplacian. We prove a quantitative stability result that goes beyond that previously obtained in [Cir+23]. Moreover, we discuss in detail several techniques and challenges in obtaining the optimal exponent in this stability result. In particular, this includes an upper bound on the exponent via an explicit computation involving a family of ellipsoids. We also sharply investigate a technique that was proposed in [Cir+18] to obtain the optimal stability exponent in the quantitative estimate for the nonlocal Alexandrov's soap bubble theorem, obtaining accurate estimates to be compared with a new, explicit example.

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arXiv2023

Remarks about the mean value property and some weighted Poincaré-type inequalities

We start providing a quantitative stability theorem for the rigidity of an overdetermined problem involving harmonic functions in a punctured domain. Our approach is inspired by and based on the proof of rigidity established by Enciso and Peralta-Salas, and reveals essential differences with respect to the stability results obtained in the literature for the classical overdetermined Serrin problem. Secondly, we provide new weighted Poincaré-type inequalities for vector fields. These are crucial tools for the study of the quantitative stability issue initiated by the author concerning a class of rigidity results involving mixed boundary value problems. Finally, we provide a mean value-type property and an associated weighted Poincaré-type inequality for harmonic functions in cones. A duality relation between this new mean value property and a partially overdetermined boundary value problem is discussed, providing an extension of a classical result due to Payne and Schaefer.

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arXiv2023

The role of antisymmetric functions in nonlocal equations

We prove a Hopf-type lemma for antisymmetric super-solutions to the Dirichlet problem for the fractional Laplacian with zero-th order terms. As an application, we use such a Hopf-type lemma in combination with the method of moving planes to prove symmetry for the semilinear fractional parallel surface problem. That is, we prove that non-negative solutions to semilinear Dirichlet problems for the fractional Laplacian in a bounded open set $Ω\subset \mathbb R^n$ must be radially symmetric if one of their level surfaces is parallel to the boundary of $Ω$; in turn, $Ω$ must be a ball. Furthermore, we discuss maximum principles and the Harnack inequality for antisymmetric functions in the fractional setting and provide counter-examples to these theorems when only `local' assumptions are imposed on the solutions.

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arXiv2022

Soap bubbles and convex cones: optimal quantitative rigidity

We consider a class of rigidity results in a convex cone $Σ\subseteq \mathbb{R}^N$. These include overdetermined Serrin-type problems for a mixed boundary value problem relative to $Σ$, Alexandrov's soap bubble-type results relative to $Σ$, and a Heintze-Karcher's inequality relative to $Σ$. Each rigidity result is obtained by means of a single integral identity and holds true under weak integral conditions. Optimal quantitative stability estimates are obtained in terms of an $L^2$-pseudodistance. In particular, the optimal stability estimate for Heintze-Karcher's inequality is new even in the classical case $Σ= \mathbb{R}^N$. Stability bounds in terms of the Hausdorff distance are also provided. Several new results are established and exploited, including a new Poincaré-type inequality for vector fields whose normal components vanish on a portion of the boundary and an explicit (possibly weighted) trace theory -- relative to the cone $Σ$ -- for harmonic functions satisfying a homogeneous Neumann condition on the portion of the boundary contained in $\partial Σ$. We also introduce new notions of uniform interior and exterior sphere conditions relative to the cone $Σ\subseteq \mathbb{R}^N$, which allow to obtain (via barrier arguments) uniform lower and upper bounds for the gradient in the mixed boundary value-setting. In the particular case $Σ= \mathbb{R}^N$, these conditions return the classical uniform interior and exterior sphere conditions (together with the associated classical gradient bounds of the Dirichlet setting).

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arXiv2022

Symmetry and quantitative stability for the parallel surface fractional torsion problem

We study symmetry and quantitative approximate symmetry for an overdetermined problem involving the fractional torsion problem in a bounded domain $Ω\subset \mathbb R^n$. More precisely, we prove that if the fractional torsion function has a $C^1$ level surface which is parallel to the boundary $\partial Ω$ then the domain is a ball. If instead we assume that the solution is close to a constant on a parallel surface to the boundary, then we quantitatively prove that $Ω$ is close to a ball. Our results use techniques which are peculiar to the nonlocal case as, for instance, quantitative versions of fractional Hopf boundary point lemma and boundary Harnack estimates for antisymmetric functions. We also provide an application to the study of rural-urban fringes in population settlements.

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arXiv2022

Radial symmetry of solutions to anisotropic and weighted diffusion equations with discontinuous nonlinearities

We prove radial symmetry for bounded nonnegative solutions of a weighted anisotropic problem. Given the anisotropic setting that we deal with, the term "radial" is understood in the Finsler framework. In the whole space, J. Serra obtained the symmetry result in the isotropic unweighted setting. In this case we provide the extension of his result to the anisotropic setting. This provides a generalization to the anisotropic setting of a celebrated result due to Gidas-Ni-Nirenberg and such a generalization is new even for in the case of linear operators whenever the dimension is greater than 2. In proper cones, the results presented are new even in the isotropic and unweighted setting for suitable nonlinear cases. Even for the previously known case of unweighted isotropic setting, the present paper provides an approach to the problem by exploiting integral (in)equalities which is new for $N>2$: this complements the corresponding symmetry result obtained via the moving planes method by Berestycki-Pacella.

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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.

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