arXiv subjects

Giulio Ciraolo

Publications and source records attributed to Giulio Ciraolo.

At least 37 records · Page 2Linked to original sources

arXiv2023

Quantitative results for fractional overdetermined problems in exterior and annular sets

We consider overdetermined problems related to the fractional capacity. In particular we study $s$-harmonic functions defined in unbounded exterior sets or in bounded annular sets, and having a level set parallel to the boundary. We first classify the solutions of the overdetermined problems, by proving that the domain and the solution itself are radially symmetric. Then we prove a quantitative stability counterpart of the symmetry results: we assume that the overdetermined condition is slightly perturbed and we measure, in a quantitative way, how much the domain is close to a symmetric set.

math.AP↗

arXiv2023

Non-existence of patterns for a class of weighted degenerate operators

A classical result by Casten-Holland and Matano asserts that constants are the only positive and stable solutions to semilinear elliptic PDEs subject to homogeneous Neumann boundary condition in bounded convex domains. In other terms, this result asserts that stable patterns do not exist in convex domains. In this paper we consider a weighted version of the Laplace operator, where the weight may be singular or degenerate at the origin, and prove the nonexistence of patterns, extending the results by Casten-Holland and Matano to general weak solutions (not necessarily stable) and under a suitable assumption on the nonlinearity and the domain. Our results exhibit some intriguing behaviour of the problem according to the weight and the geometry of the domain. Indeed, our main results follow from a geometric assumption on the second fundamental form of the boundary in terms of a parameter which describes the degeneracy of the operator. As a consequence, we provide some examples and show that nonexistence of patterns may occurs also for non convex domains whenever the weight is degenerate.

math.AP↗

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.

math.AP↗

arXiv2021

Interior regularity results for inhomogeneous anisotropic quasilinear equations

We consider inhomogeneous $p$-Laplace type equations of the form $-\mathrm{div}\left(a(\nabla u)\right)=f$ in a possibly anisotropic setting. Under general assumptions on the source term $f$, we obtain quantitative Sobolev regularity results for the stress field $a(\nabla u)$ and weighted $L^2$ estimates for the Hessian of the solution. As far as we know, our results are new or refine the ones available in literature also when restricted to the Euclidean setting.

math.AP↗

arXiv2021

An exterior overdetermined problem for Finsler $N$-laplacian in convex cones

We consider a partially overdetermined problem for anisotropic $N$-Laplace equations in a convex cone $Σ$ intersected with the exterior of a bounded domain $Ω$ in $\mathbb{R}^N$, $N\geq 2$. Under a prescribed logarithmic condition at infinity, we prove a rigidity result by showing that the existence of a solution implies that $Σ\capΩ$ must be the intersection of the Wulff shape and $Σ$. Our approach is based on a Pohozaev-type identity and the characterization of minimizers of the anisotropic isoperimetric inequality inside convex cones.

math.AP↗

arXiv2021

Symmetry for positive critical points of Caffarelli-Kohn-Nirenberg inequalities

We consider positive critical points of Caffarelli-Kohn-Nirenberg inequalities and prove a Liouville type result which allows us to give a complete classification of the solutions in a certain range of parameters, providing a symmetry result for positive solutions. The governing operator is a weighted $p$-Laplace operator, which we consider for a general $p \in (1,d)$. For $p=2$, the symmetry breaking region for extremals of Caffarelli-Kohn-Nirenberg inequalities was completely characterized in [J. Dolbeault, M. Esteban, M. Loss, Invent. Math. 44 (2016)]. Our results extend this result to a general $p$ and are optimal in some cases.

math.AP↗

arXiv2020

An overdetermined problem associated to the Finsler Laplacian

We prove a rigidity result for the anisotropic Laplacian. More precisely, the domain of the problem is bounded by an unknown surface supporting a Dirichlet condition together with a Neumann-type condition which is not translation-invariant. Using a comparison argument, we show that the domain is in fact a Wulff shape. We also consider the more general case when the unknown surface is required to have its boundary on a given conical surface: in such a case, the domain of the problem is bounded by the unknown surface and by a portion of the given conical surface, which supports a homogeneous Neumann condition. We prove that the unknown surface lies on the boundary of a Wulff shape.

math.AP↗

arXiv2020

Classification and non-existence results for weak solutions to quasilinear elliptic equations with Neumann or Robin boundary conditions

We classify positive solutions to a class of quasilinear equations with Neumann or Robin boundary conditions in convex domains. Our main tool is an integral formula involving the trace of some relevant quantities for the problem. Under a suitable condition on the nonlinearity, a relevant consequence of our results is that we can extend to weak solutions a celebrated result obtained for stable solutions by Casten and Holland and by Matano.

math.AP↗

arXiv2020

Quantitative estimates for almost constant mean curvature hypersurfaces

Alexandrov's soap bubble theorem asserts that spheres are the only connected closed embedded hypersurfaces in the Euclidean space with constant mean curvature. The theorem can be extended to space forms and it holds for more general functions of the principal curvatures. In this short review, we discuss quantitative stability results regarding Alexandrov's theorem which have been obtained by the author in recent years. In particular, we consider hypersurfaces having mean curvature close to a constant and we quantitatively describe the proximity to a single sphere or to a collection of tangent spheres in terms of the oscillation of the mean curvature. Moreover, we also consider the problem in a non local setting, and we show that the non local effect gives a stronger rigidity to the problem and prevents the appearance of bubbling.

math.AP↗

arXiv2019

Symmetry results for critical anisotropic $p$-Laplacian equations in convex cones

Given $n \geq 2$ and $1<p<n$, we consider the critical $p$-Laplacian equation $Δ_p u + u^{p^*-1}=0$, which corresponds to critical points of the Sobolev inequality. Exploiting the moving planes method, it has been recently shown that positive solutions in the whole space are classified. Since the moving plane method strongly relies on the symmetries of the equation and the domain, in this paper we provide a new approach to this Liouville-type problem that allows us to give a complete classification of solutions in an anisotropic setting. More precisely, we characterize solutions to the critical $p$-Laplacian equation induced by a smooth norm inside any convex cone. In addition, using optimal transport, we prove a general class of (weighted) anisotropic Sobolev inequalities inside arbitrary convex cones.

math.AP↗

arXiv2018

Quantitative stability for hypersurfaces with almost constant curvature in space forms

The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures $f(k_1,\ldots,k_{n-1})$ satisfying suitable conditions. In this paper we give sharp quantitative estimates of proximity to a single sphere for Alexandrov Soap Bubble Theorem in space forms when the curvature operator $f$ is close to a constant. Under an assumption that prevents bubbling, the proximity to a single sphere is quantified in terms of the oscillation of the curvature function $f$. Our approach provides a unified picture of quantitative studies of the method of moving planes in space forms.

math.DG↗

arXiv2018

The method of moving planes: a quantitative approach

We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric analysis (like Alexandrov soap bubble Theorem), and we give an overview of some recent results related to quantitative studies of the method of moving planes, where quantitative approximate symmetry results are obtained.

math.AP↗

arXiv2018

Stress concentration for closely located inclusions in nonlinear perfect conductivity problems

We study the stress concentration, which is the gradient of the solution, when two smooth inclusions are closely located in a possibly anisotropic medium. The governing equation may be degenerate of $p-$Laplace type, with $1<p \leq N$. We prove optimal $L^\infty$ estimates for the blow-up of the gradient of the solution as the distance between the inclusions tends to zero.

math.AP↗

arXiv2018

Quantitative stability for hypersurfaces with almost constant mean curvature in the hyperbolic space

We provide sharp stability estimates for the Alexandrov Soap Bubble Theorem in the hyperbolic space. The closeness to a single sphere is quantified in terms of the dimension, the measure of the hypersurface and the radius of the touching ball condition. As consequence we obtain a new pinching result for hypersurfaces in the hyperbolic space. Our approach is based on the method of moving planes. In this context we carefully review the method and we provide the first quantitative study in the hyperbolic space.

math.DG↗

arXiv2018

Serrin's type overdetermined problems in convex cones

We consider overdetermined problems of Serrin's type in convex cones for (possibly) degenerate operators in the Euclidean space as well as for a suitable generalization to space forms. We prove rigidity results by showing that the existence of a solution implies that the domain is a spherical sector.

math.AP↗

arXiv2018

A remark on constant mean curvature hypersurfaces in warped product manifolds

Alexandrov's theorem asserts that spheres are the only closed embedded constant mean curvature hypersurfaces in space forms. In this paper, we consider Alexandrov's theorem in warped product manifolds and prove a rigidity result in the spirit of Alexandrov's theorem. Our approach generalizes the proofs of Reilly and Ros and, under more restrictive assumptions, it provides an alternative proof of a recent theorem of Brendle.

math.AP↗

arXiv2018

Some overdetermined problems related to the anisotropic capacity

We characterize the Wulff shape of an anisotropic norm in terms of solutions to overdetermined problems for the Finsler $p$-capacity of a convex set $Ω\subset \mathbb{R}^N$, with $1<p<N$. In particular we show that if the Finsler $p$-capacitary potential $u$ associated to $Ω$ has two homothetic level sets then $Ω$ is Wulff shape. Moreover, we show that the concavity exponent of $u$ is $q=-(p-1)/(N-p)$ if and only if $Ω$ is Wulff shape.

math.AP↗

arXiv2018

Gradient estimates for the perfect conductivity problem in anisotropic media

We study the perfect conductivity problem when two perfectly conducting inclusions are closely located to each other in an anisotropic background medium. We establish optimal upper and lower gradient bounds for the solution in any dimension which characterize the singular behavior of the electric field as the distance between the inclusions goes to zero.

math.AP↗