arXiv subjects

Fabio De Regibus

Publications and source records attributed to Fabio De Regibus.

8 recordsLinked to original sources

arXiv2026

Bifurcations in Isoperimetric Problems with Nonlocal Interactions

We study isoperimetric problems modeled on the liquid drop model, with nonlocal interactions under a volume constraint. While balls are natural critical points, we show that, for an unbounded sequence of radii, non-spherical solutions bifurcate from the family of balls. These new solutions lie arbitrarily close to balls and can have arbitrarily large volume. Conversely, at radii outside this sequence, no bifurcation occurs, and nearby solutions are trivial, arising only from rigid motions.

math.AP↗

arXiv2025

Rigidity results for finite energy solutions to the stationary 2D Euler equations

In this paper we prove rigidity results for classical solutions to the stationary 2D Euler equations in $\mathbb{R}^2$. Assuming that the velocity field has finite energy and that the stagnation set is connected, we prove that the corresponding stream function solves an autonomous semilinear elliptic equation. Under some extra conditions on the vorticity near infinity we can also prove that the streamlines are concentric circles. The proofs include several energy estimates on the behavior of the stream function at infinity, as well as an adaptation of the continuous Steiner symmetrization to our setting.

math.AP↗

arXiv2024

On the critical points of solutions of Robin boundary problems

In this paper we prove the uniqueness of the critical point for stable solutions of the Robin problem \[ \begin{cases} -Δu=f(u)&\text{in }Ω\\ u>0&\text{in }Ω\\ \partial_νu+βu=0&\text{on }\partialΩ, \end{cases} \] where $Ω\subseteq\mathbb{R}^2$ is a smooth and bounded domain with strictly positive curvature of the boundary, $f\ge0$ is a smooth function and $β>0$. Moreover, for $β$ large the result fails as soon as the domain is no more convex, even if it is very close to be: indeed, in this case it is possible to find solutions with an arbitrary large number of critical points.

math.AP↗

arXiv2024

Monotone heteroclinic solutions to semilinear PDEs in cylinders and applications

In this paper we show the existence of strictly monotone heteroclinic type solutions of semilinear elliptic equations in cylinders. The motivation of this construction is twofold: first, it implies the existence of an entire bounded solution of a semilinear equation without critical points which is not one-dimensional. Second, this gives an example of a bounded stationary solution for the 2D Euler equations without stagnation points which is not a shear flow, completing previous results of Hamel and Nadirashvili. The proof uses a minimization technique together with a truncation argument, and a limit procedure.

math.AP↗

arXiv2023

On the shape of solutions to elliptic equations in possibly non convex planar domains

In this note we prove uniqueness of the critical point for positive solutions of elliptic problems in bounded planar domains: we first examine the Poisson problem - Delta u = f(x,y) finding a geometric condition involving the curvature of the boundary and the normal derivative of f on the boundary to ensure uniqueness of the critical point. In the second part we consider stable solutions of the nonlinear problem -Delta u = f(u) in perturbation of convex domains.

math.AP↗

arXiv2021

On the number of critical points of the second eigenfunction of the Laplacian in convex planar domains

In this paper we consider the second eigenfunction of the Laplacian with Dirichlet boundary conditions in convex domains. If the domain has \emph{large eccentricity} then the eigenfunction has \emph{exactly} two nondegenerate critical points (of course they are one maximum and one minimum). The proof uses some estimates proved by Jerison ([Jer95a]) and Grieser-Jerison ([GJ96]) jointly with a topological degree argument. Analogous results for higher order eigenfunctions are proved in rectangular-like domains considered in [GJ09].

math.AP↗

arXiv2021

On the number of critical points of stable solutions in bounded strip-like domains

In this paper we show that there exists a family of domains $Ω_{\varepsilon}\subseteq\mathbb{R}^N$ with $N\ge2$, such that the $stable$ solution of the problem \[ \begin{cases} -Δu= g(u)&\hbox{in }Ω_\varepsilon\\ u>0&\hbox{in }Ω_\varepsilon\\ u=0&\hbox{on }\partialΩ_\varepsilon \end{cases} \] admits $k$ critical points with $k\ge2$. Moreover the sets $Ω_\varepsilon's$ are star-shaped and "close" to a strip as $\varepsilon\to0$. Next, if $g(u)\equiv1$ and $N\ge3$ we exhibit a family of domain $Ω_\varepsilon's$ with $positive$ $mean$ $curvature$ and solutions $u_\varepsilon $ which have $k$ critical points with $k\ge2$. In this case, the domains $Ω_\varepsilon $ turn out to be "close" to a cylinder as $\varepsilon\to0$.

math.AP↗

arXiv2020

Uniqueness of the critical point for semi-stable solution in $\mathbb{R}^2$

In this paper we show the uniqueness of the critical point for \emph{semi-stable} solutions of the problem $$\begin{cases} -Δu=f(u)&\text{in }Ω\\ u>0&\text{in }Ω\\ u=0&\text{on } \partialΩ,\end{cases}$$ where $Ω\subset\mathbb{R}^2$ is a smooth bounded domain whose boundary has \emph{nonnegative} curvature and $f(0)\ge0$. It extends a result by Cabré-Chanillo to the case where the curvature of $\partialΩ$ vanishes.

math.AP↗