arXiv subjects

Giorgio Tortone

Publications and source records attributed to Giorgio Tortone.

At least 19 recordsLinked to original sources

arXiv2026

Optimal partition and segregation problems driven by torsional rigidity

Spectral optimal partition and segregation problems are deeply connected with harmonic maps, eigenfunctions, and the fine structure of nodal sets for linear elliptic equations. In this paper, we show that replacing the spectral energy by torsional rigidity leads to a genuinely different theory. The resulting optimal configurations are governed locally not by harmonic equations, but by torsion-type energies and unstable free boundary problems, thereby creating a natural bridge between optimal partition theory and the analysis of sublinear free boundary phenomena. We prove existence of optimal torsional partitions and segregated torsional configurations, together with optimal Lipschitz regularity of the associated nonlinear eigenfunctions. We establish a strong unique continuation principle, characterize the admissible vanishing orders and the corresponding blow-up profiles, derive sharp Hausdorff dimension estimates for the nodal set and its singular subset, and prove $C^{1,\alpha}$-regularity of the regular part of the free boundary. The proofs combine variational arguments with Almgren-type and Weiss-type monotonicity formulae adapted to the intrinsically sublinear torsional regime, blow-up analysis, and tools from geometric measure theory.

math.AP↗

arXiv2026

Uniqueness of the blow-up for some Alt-Phillips cones

We establish uniqueness of blow-ups, with sharp quantitative convergence, for several classes of singular minimizing cones in the Alt-Phillips problem, in the range $\gamma\in(0,2)$. As a consequence, we obtain uniqueness at every free boundary point in dimensions $d=2,3,4$ for $\gamma\in(1,2)$, and in dimensions $d\geq 5$ for $\gamma\in\left(1,\frac32\right)$. The proof of uniqueness is based on three new logarithmic epiperimetric inequalities. The sharp distinction between polynomial and logarithmic convergence is governed by a finite-dimensional integrability condition (sub-integrability) for the spherical linearized problem. We prove this sharpness for the radial cone and its cylindrical extensions through an explicit integrability and bifurcation analysis, showing in particular that logarithmic convergence may be sharp even in dimension two. In contrast, we show that the one-dimensional cone is exceptional: although the integrability condition fails, the convergence is polynomial. Finally, we characterize the minimality of the radial cone in terms of $d$ and $\gamma$ by means of a one-dimensional calibration argument, exhibiting in dimension $d\geq6$ a nontrivial regime in which the radial cone is stable but not minimizing.

math.AP↗

arXiv2025

Vectorial Bernoulli Problems and Free Boundary Systems

In this survey we go through some of the recent results about the regularity of vectorial free boundary problems of Bernoulli type and free boundary systems. The aim is to illustrate the general methodologies as well as to outline a selection of notable open questions.

math.AP↗

arXiv2025

A priori H\"older estimates for equations degenerating on nodal sets

We prove a priori H\"older bounds for continuous solutions to degenerate equations with variable coefficients of type $$ \mathrm{div}\left(u^2 A\nabla w\right)=0\quad\mathrm{in \ }\Omega\subset\mathbb R^n,\qquad \mbox{with}\qquad \mathrm{div}\left(A\nabla u\right)=0, $$ where $A$ is a Lipschitz continuous, uniformly elliptic matrix (possibly $u$ has non-trivial singular nodal set). Such estimates are uniform with respect to $u$ in a class of normalized solutions that have a bounded Almgren frequency. As a consequence, a boundary Harnack principle holds for the quotient of two solutions vanishing on a common set. This analysis relies on a detailed study of the associated weighted Sobolev spaces, including integrability of the weight, capacitary properties of the nodal set, and uniform Sobolev inequalities yielding local boundedness of solutions.

math.AP↗

arXiv2025

Smoothness and stability in the Alt-Phillips problem

We study the one-phase Alt-Phillips free boundary problem, focusing on the case of negative exponents $\gamma \in (-2,0)$. The goal of this paper is twofold. On the one hand, we prove smoothness of $C^{1,\alpha}$-regular free boundaries by reducing the problem to a class of degenerate quasilinear PDEs, for which we establish Schauder estimates. Such method provide a unified proof of the smoothness for general exponents. On the other hand, by exploiting the higher regularity of solutions, we derive a new stability condition for the Alt-Phillips problem in the negative exponent regime, ruling out the existence of nontrivial axially symmetric stable cones in low dimensions. Finally, we provide a variational criterion for the stability of cones in the Alt-Phillips problem, which recovers the one for minimal surfaces in the singular limit as $\gamma \to -2$.

math.AP↗

arXiv2025

Some remarks on singular capillary cones with free boundary

We study minimizing singular cones with free boundary associated with the capillarity problem. Precisely, we provide a stability criterion $\`a$ la Jerison-Savin for capillary hypersurfaces and show that, in dimensions up to $4$, minimizing cones with non-sign-changing mean curvature are flat. We apply this criterion to minimizing capillary drops and, additionally, establish the instability of non-trivial axially symmetric cones in dimensions up to $6$. The main results are based on a Simons-type inequality for a class of convex, homogeneous, symmetric functions of the principal curvatures, combined with a boundary condition specific to the capillary setting.

math.DG↗

arXiv2024

A priori regularity estimates for equations degenerating on nodal sets

We prove a priori and a posteriori H\"older bounds and Schauder $C^{1,\alpha}$ estimates for continuous solutions of degenerate elliptic equations with variable coefficients of the form $$ \mathrm{div}\left(|u|^a A\nabla w\right)=0\qquad\mathrm{in \ }\Omega\subset\mathbb R^2,\quad a\in\mathbb R, $$ where the weight $u$ is itself a solution to an elliptic equation of the type $\mathrm{div}(A \nabla u) = 0$, with $A$ a Lipschitz-continuous, uniformly elliptic matrix. The function $u$ is allowed to have a nontrivial, possibly singular nodal set. The estimates are uniform with respect to $u$ within a class of normalized solutions having bounded Almgren frequency. In the special case $a = 2$, our results apply to the ratio of two solutions to the same elliptic equation sharing a common zero set. Precisely, we prove higher-order boundary Harnack principles on nodal domains, via the derived Schauder estimates for the associated degenerate equations. The results are based upon a fine blow-up argument, a Liouville theorem, and quasiconformal maps.

math.AP↗

arXiv2024

On the nodal set of solutions to some sublinear equations without homogeneity

We investigate the structure of the nodal set of solutions to an unstable Alt-Phillips type problem \[ -Δu = λ_+(u^+)^{p-1}-λ_-(u^-)^{q-1} \] where $1 \le p 0$, $λ_- \ge 0$. The equation is characterized by the sublinear inhomogeneous character of the right hand-side, which makes difficult to adapt in a standard way classical tools from free-boundary problems, such as monotonicity formulas and blow-up arguments. Our main results are: the local behavior of solutions close to the nodal set; the complete classification of the admissible vanishing orders, and estimates on the Hausdorff dimension of the singular set, for local minimizers; the existence of degenerate (not locally minimal) solutions.

math.AP↗

arXiv2024

Higher order boundary Harnack principle via degenerate equations

As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type: \[ -\mathrm{div}\left(ρ^aA\nabla w\right)=ρ^af+\mathrm{div}\left(ρ^aF\right) \quad\textrm{in}\; Ω\] for exponents $a>-1$, where the weight $ρ$ vanishes in a non degenerate manner on a regular hypersurface $Γ$ which can be either a part of the boundary of $Ω$ or mostly contained in its interior. As an application, we extend such estimates to the ratio $v/u$ of two solutions to a second order elliptic equation in divergence form when the zero set of $v$ includes the zero set of $u$ which is not singular in the domain (in this case $ρ=u$, $a=2$ and $w=v/u$). We prove first $C^{k,α}$-regularity of the ratio from one side of the regular part of the nodal set of $u$ in the spirit of the higher order boundary Harnack principle established by De Silva and Savin. Then, by a gluing Lemma, the estimates extend across the regular part of the nodal set. Finally, using conformal mapping in dimension $n=2$, we provide local gradient estimates for the ratio which hold also across the singular set.

math.AP↗

arXiv2024

Liouville theorems and optimal regularity in elliptic equations

The objective of this paper is to establish a connection between the problem of optimal regularity among solutions to elliptic PDEs with measurable coefficients and the Liouville property at infinity. Initially, we address the two-dimensional case by proving an Alt-Caffarelli-Friedman type monotonicity formula, enabling the proof of optimal regularity and the Liouville property for multiphase problems. In higher dimensions, we delve into the role of monotonicity formulas in characterizing optimal regularity. By employing a hole-filling technique, we present a distinct "almost-monotonicity" formula that implies H$ö$lder regularity of solutions. Finally, we explore the interplay between the least growth at infinity and the exponent of regularity by combining blow-up and $G$-convergence arguments.

math.AP↗

arXiv2024

The boundary Harnack principle on optimal domains

We give a short and self-contained proof of the Boundary Harnack inequality for a class of domains satisfying some geometric conditions given in terms of a state function that behaves as the distance function to the boundary, is subharmonic inside the domain and satisfies some suitable estimates on the measure of its level sets. We also discuss the applications of this result to some shape optimization and free boundary problems.

math.AP↗

arXiv2023

A capillarity one-phase Bernoulli free boundary problem

We consider a one-phase Bernoulli free boundary problem in a container $D$ - a smooth open subset of $\mathbb{R}^d$ - under the condition that on the fixed boundary $\partial D$ the normal derivative of the solutions is prescribed. We study the regularity of the free boundary (the boundary of the positivity set of the solution) up to $\partial D$ and the structure of the wetting region, which is the contact set between the free boundary and the ($(d-1)$-dimensional) fixed boundary $\partial D$. In particular, we characterize the contact angle in terms of the permeability of the porous container and we show that the boundary of the wetting region is a smooth $(d-2)$-dimensional manifold, up to a (possibly empty) closed set of Hausdorff dimension at most $d-5$.

math.AP↗

arXiv2023

On the dimension of the singular set in optimization problems with measure constraint

In this paper, we prove estimates on the dimension of the singular part of the free boundary for solutions to shape optimization problems with measure constraints. The focus is on the heat conduction problem studied by Aguilera, Caffarelli, and Spruck and the one-phase Bernoulli problem with measure constraint introduced by Aguilera, Alt and Caffarelli. To estimate the Hausdorff dimension of the singular set, we introduce a new formulation of the notion of stability for the one-phase problem along volume-preserving variations, which is preserved under blow-up limits. Finally, the result follows by applying the program developed in [Buttazzo et al. 2022] to this class of domain variation.

math.AP↗

arXiv2022

Regularity of the optimal sets for a class of integral shape functionals

We prove {the first} regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain $Ω$ is obtained as the integral of a cost function $j(u,x)$ depending on the solution $u$ of a certain PDE problem on $Ω$. The main feature of these functionals is that the minimality of a domain $Ω$ cannot be translated into a variational problem for a single (real or vector valued) state function. In this paper we focus on the case of affine cost functions $j(u,x)=-g(x)u+Q(x)$, where $u$ is the solution of the PDE $-Δu=f$ with Dirichlet boundary conditions. We obtain the Lipschitz continuity and the non-degeneracy of the optimal $u$ from the inwards/outwards optimality of $Ω$ and then we use the stability of $Ω$ with respect to variations with smooth vector fields in order to study the blow-up limits of the state function $u$. By performing a triple consecutive blow-up, we prove the existence of blow-up sequences converging to homogeneous stable solution of the one-phase Bernoulli problem and according to the blow-up limits, we decompose $\partialΩ$ into a singular and a regular part. In order to estimate the Hausdorff dimension of the singular set of $\partialΩ$ we give a new formulation of the notion of stability for the one-phase problem, which is preserved under blow-up limits and allows to develop a dimension reduction principle. Finally, by combining a higher order Boundary Harnack principle and a viscosity approach, we prove $C^\infty$ regularity of the regular part of the free boundary when the data are smooth.

math.AP↗

arXiv2021

Regularity of shape optimizers for some spectral fractional problems

This paper is dedicated to the spectral optimization problem $$ \mathrm{min}\left\{λ_1^s(Ω)+\cdots+λ_m^s(Ω) + Λ\mathcal{L}_n(Ω)\colon Ω\subset D \mbox{ s-quasi-open}\right\} $$ where $Λ>0, D\subset \mathbb{R}^n$ is a bounded open set and $λ_i^s(Ω)$ is the $i$-th eigenvalues of the fractional Laplacian on $Ω$ with Dirichlet boundary condition on $\mathbb{R}^n\setminus Ω$. We first prove that the first $m$ eigenfunctions on an optimal set are locally Hölder continuous in the class $C^{0,s}$ and, as a consequence, that the optimal sets are open sets. Then, via a blow-up analysis based on a Weiss type monotonicity formula, we prove that the topological boundary of a minimizer $Ω$ is composed of a relatively open regular part and a closed singular part of Hausdorff dimension at most $n-n^*$, for some $n^*\geq 3$. Finally we use a viscosity approach to prove $C^{1,α}$-regularity of the regular part of the boundary.

math.AP↗

arXiv2021

Epsilon-regularity for the solutions of a free boundary system

This paper is dedicated to a free boundary system arising in the study of a class of shape optimization problems. The problem involves three variables: two functions $u$ and $v$, and a domain $Ω$; with $u$ and $v$ being both positive in $Ω$, vanishing simultaneously on $\partialΩ$ and satisfying an overdetermined boundary value problem involving the product of their normal derivatives on $\partialΩ$. Precisely, we consider solutions $u, v \in C(B_1)$ of $$-Δu= f \quad\text{and} \quad-Δv=g\quad\text{in}\quad Ω=\{u>0\}=\{v>0\}\ ,\qquad \frac{\partial u}{\partial n}\frac{\partial v}{\partial n}=Q\quad\text{on}\quad \partialΩ\cap B_1.$$ Our main result is an epsilon-regularity theorem for viscosity solutions of this free boundary system. We prove a partial Harnack inequality near flat points for the couple of auxiliary functions $\sqrt{uv}$ and $\frac12(u+v)$. Then, we use the gained space near the free boundary to transfer the improved flatness to the original solutions. Finally, using the partial Harnack inequality, we obtain an improvement-of-flatness result, which allows to conclude that flatness implies $C^{1,α}$ regularity.

math.AP↗

arXiv2020

A vectorial problem with thin free boundary

We consider the vectorial analogue of the thin free boundary problem introduced in \cite{CRS} as a realization of a nonlocal version of the classical Bernoulli problem. We study optimal regularity, nondegeneracy, and density properties of local minimizers. Via a blow-up analysis based on a Weiss type monotonicity formula, we show that the free boundary is the union of a "regular" and a "singular" part. Finally we use a viscosity approach to prove $C^{1,α}$ regularity of the regular part of the free boundary.

math.AP↗

arXiv2020

The nodal set of solutions to some nonlocal sublinear problems

We study the nodal set of solutions to equations of the form $$ (-Δ)^s u = λ_+ (u_+)^{q-1} - λ_- (u_-)^{q-1}\quad\text{in $B_1$}, $$ where $λ_+,λ_->0, q \in [1,2)$, and $u_+$ and $u_-$ are respectively the positive and negative part of $u$. This collection of nonlinearities includes the unstable two-phase membrane problem $q=1$ as well as sublinear equations for $1<q<2$. We initially prove the validity of the strong unique continuation property and the finiteness of the vanishing order, in order to implement a blow-up analysis of the nodal set. As in the local case $s=1$, we prove that the admissible vanishing orders can not exceed the critical value $k_q= 2s/(2- q)$. Moreover, we study the regularity of the nodal set and we prove a stratification result. Ultimately, for those parameters such that $k_q< 1$, we prove a remarkable difference with the local case: solutions can only vanish with order $k_q$ and the problem admits one dimensional solutions. Our approach is based on the validity of either a family of Almgren-type or a 2-parameter family of Weiss-type monotonicity formulas, according to the vanishing order of the solution.

math.AP↗