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Danilo Gregorin Afonso

Publications and source records attributed to Danilo Gregorin Afonso.

8 recordsLinked to original sources
arXiv2026

Non-radial minimizers for the first eigenvalue of the Laplacian in cones and a related overdetermined problem

In this work, we consider relative overdetermined problems for the first eigenfunction of the Laplacian for domains in cones, and the related question of minimizing the first eigenvalue among sets of a given fixed measure. By means of a shape derivative analysis, we show that the spherical sector is a critical shape and obtain a geometric condition on the cone for its stability/instability. By a concentration-compactness argument, we prove the existence of a minimizer, which moreover is bounded, open, connected, and whose relative boundary is regular almost everywhere. By another domain variation argument, we conclude that the minimizers admit a solution for the overdetermined problem.

math.AP↗
arXiv2025

Existence of solutions with prescribed frequency for the perturbed Schrödinger-Bopp-Podolsky system in bounded domains

In this paper, we show that the Schrödinger-Bopp-Podolsky system with Dirichlet boundary conditions in a bounded domain possesses infinitely many solutions of prescribed frequency, for any set of (continuous) boundary conditions, provided that the Schrödinger equation is perturbed with a suitable nonlinearity. Our approach is variational, and our proof is based on a symmetric variant of the Mountain Pass theorem.

math.AP↗
arXiv2025

Multiple solutions to asymptotically linear problems driven by superposition operators

In this paper, we investigate the existence and multiplicity of weak solutions to problems involving a superposition operator of the type $$\int_{[0, 1]}(- Δ)^s u d μ(s),$$ for a signed measure $μ$ on the interval of fractional exponents $[0,1]$, when the nonlinearity is subcritical and asymptotically linear at infinity; thus, we deal with a perturbation of the eigenvalue problem for the superposition operator. We use variational tools, extending to this setting well-known results for the classical and the fractional Laplace operators.

math.AP↗
arXiv2024

Semilinear equations in bounded cylinders: Morse index and bifurcation from one-dimensional solutions

In this paper, we study semilinear elliptic equations in domains where there is a natural class of solutions, which depend only on one variable, and whose simple geometry reflects the geometry of the domain. We prove that under quite general assumptions, other types of solutions also exist. More precisely, we consider one-dimensional solutions in bounded cylinders and, combining a suitable separation of variables with the theory of ordinary differential equations, we show how to compute the Morse index of such solutions. The Morse index is then used to prove local and global bifurcation results.

math.AP↗
arXiv2023

Energy stability for a class of semilinear elliptic problems

In this paper, we consider semilinear elliptic problems in a bounded domain $Ω$ contained in a given unbounded Lipschitz domain $\mathcal C \subset \mathbb R^N$. Our aim is to study how the energy of a solution behaves with respect to volume-preserving variations of the domain $Ω$ inside $\mathcal C$. Once a rigorous variational approach to this question is set, we focus on the cases when $\mathcal C$ is a cone or a cylinder and we consider spherical sectors and radial solutions or bounded cylinders and special one-dimensional solutions, respectively. In these cases, we show both stability and instability results, which have connections with related overdetermined problems.

math.AP↗
arXiv2022

Overdetermined problems and relative Cheeger sets in unbounded domains

In this paper we study a partially overdetermined mixed boundary value problem for domains $Ω$ contained in an unbounded set $\mathcal C$. We introduce the notion of Cheeger set relative to $\mathcal C$ and show that if a domain $Ω\subset \mathcal C$ admits a solution of the overdetermined problem, then it coincides with its relative Cheeger set. We also study the related problem of characterizing constant mean curvature surfaces $Γ$ inside $\mathcal C$. In the case when $\mathcal C$ is a cylinder we obtain further results whenever the relative boundary of $Ω$ or the surface $Γ$ is a graph on the base of the cylinder.

math.AP↗
arXiv2020

Intercultural Science-Art Project 'Magritte Meet Science'

In this manuscript a joint project between young researchers from all around the world, the Intercultural Science-Art Project, is presented, in which all participants interpreted their own research with art. Art is meant in the most inclusive way possible, including drawings and digital works, but also songs and confectionary art. This project has two main goals. On the one hand it creates tools for young scientists to communicate their research to society by artistic means. On the other hand it tries to bring together different cultures and to strengthen ties between them.

math.HO↗