arXiv subjects

Matteo Carducci

Publications and source records attributed to Matteo Carducci.

8 recordsLinked to original sources

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

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

Free boundary regularity for a tumor growth model with obstacle

We develop an existence and regularity theory for solutions to a geometric free boundary problem motivated by models of tumor growth. In this setting, the tumor invades an accessible region $D$, its motion is directed along a constant vector $V$, and it cannot penetrate another region $K$ acting as an obstacle to the spread of the tumor. Due to the non variational structure of the problem, we show existence of viscosity solutions via Perron's method. Subsequently, we prove interior regularity for the free boundary near regular points by means of an improvement of flatness argument. We further analyze the boundary regularity and we prove that the free boundary meets the obstacle as a $C^{1,\alpha}$ graph. A key step in the analysis of the boundary regularity involves the study of a thin obstacle problem with oblique boundary conditions, for which we establish $C^{1,\alpha}$ estimates.

math.AP↗

arXiv2025

Existence and regularity in the fully nonlinear one-phase free boundary problem

We consider viscosity solution to one-phase free boundary problems for general fully nonlinear operators and free boundary condition depending on the normal vector. We show existence of viscosity solutions via the Perron's method and we prove $C^{2,\alpha}$ regularity of flat free boundaries via a quadratic improvement of flatness. Finally, we obtain the higher regularity of the free boundary via an hodograph transform.

math.AP↗

arXiv2024

Generic regularity of free boundaries in the obstacle problem for the fractional Laplacian

We establish generic regularity results of free boundaries for solutions of the obstacle problem for the fractional Laplacian $(-\Delta)^s$. We prove that, for almost every obstacle, the free boundary contains only regular points up to dimension $3$, for every $s\in(0,1)$. To do so, we extend some results on the fine structure of the free boundary to the case $s\in (0,1)$ and general non-zero obstacle, including a blow-up analysis at points with frequency $2m+2s$, and we prove new explicit uniform frequency gaps for solutions of the fractional obstacle problem.

math.AP↗

arXiv2024

An epiperimetric inequality for odd frequencies in the thin obstacle problem

We prove for the first time an epiperimetric inequality for the thin obstacle Weiss' energy with odd frequencies and we apply it to solutions to the thin obstacle problem with general $C^{k,\gamma}$. In particular, we obtain the rate of convergence of the blow-up sequences at points of odd frequencies and the regularity of the strata of the corresponding contact set. We also recover the frequency gap for odd frequencies obtained by Savin and Yu.

math.AP↗

arXiv2023

Epiperimetric inequalities in the obstacle problem for the fractional Laplacian

Using the epiperimetric inequalities approach, we study the obstacle problem $\min\{(-\Delta)^su,u-\varphi\}=0,$ for the fractional Laplacian $(-\Delta)^s$ with obstacle $\varphi\in C^{k,\gamma}(\mathbb{R}^n)$, $k\ge2$ and $\gamma\in(0,1)$. We prove an epiperimetric inequality for the Weiss' energy $W_{1+s}$ and a logarithmic epiperimetric inequality for the Weiss' energy $W_{2m}$. Moreover, we also prove two epiperimetric inequalities for negative energies $W_{1+s}$ and $W_{2m}$. By these epiperimetric inequalities, we deduce a frequency gap and a characterization of the blow-ups for the frequencies $\lambda=1+s$ and $\lambda=2m$. Finally, we give an alternative proof of the regularity of the points on the free boundary with frequency $1+s$ and we describe the structure of the points on the free boundary with frequency $2m$, with $m\in\mathbb{N}$ and $2m\le k.$

math.AP↗

arXiv2023

Optimal regularity of the thin obstacle problem by an epiperimetric inequality

The key point to prove the optimal $C^{1,\frac12}$ regularity of the thin obstacle problem is that the frequency at a point of the free boundary $x_0\in\Gamma(u)$, say $N^{x_0}(0^+,u)$, satisfies the lower bound $N^{x_0}(0^+,u)\ge\frac32$. In this paper we show an alternative method to prove this estimate, using an epiperimetric inequality for negative energies $W_\frac32$. It allows to say that there are not $\lambda-$homogeneous global solutions with $\lambda\in (1,\frac32)$, and by this frequancy gap, we obtain the desired lower bound, thus a new self contained proof of the optimal regularity.

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