arXiv subjects

Diego A. Marín

Publications and source records attributed to Diego A. Marín.

5 recordsLinked to original sources

arXiv2026

Comparison methods for semilinear elliptic problems on Riemannian manifolds with a Ricci lower bound

In the first part of the article we develop a comparison method for positive solutions of the semilinear Dirichlet problem $Δu+f(u)=0$ on domains $Ω\subset \mathcal M^n$ of a Riemannian manifold $(\mathcal{M}^n,g)$ with a Ricci lower bound $\operatorname{Ric}_g\ge (n-1)k\,g$. Assuming admissibility and structural conditions on $f$, we prove a sharp pointwise gradient comparison, with a rigid characterization of the equality case. As applications, we derive an explicit isoperimetric-type inequality and a quantitative hot-spot localization estimate under natural convexity assumptions. In the second part, on $\mathbb S^n$ we show that isoparametric foliations produce non-rotational $f$-extremal domains, and that these examples descend to smooth quotients under free isometric actions preserving the foliation.

math.DG↗

arXiv2026

Extremal domains in $\mathbb{S}^2$: Geometric and Analytic methods

In this article, we study domains $Ω\subset \mathbb{S}^2$ that support positive solutions of the overdetermined problem $$ Δu + f(u,|\nabla u|)=0 \quad \text{in } Ω, $$ subject to the boundary conditions $u=0$ on $\partialΩ$ and $|\nabla u|$ being locally constant along $\partialΩ$. We refer to such domains as $f$--extremal domains. In the first part of the paper, we extend the moving plane method in $\mathbb{S}^2$ and show that if an $f$--extremal domain $Ω$ contains a simple curve of maximum points of $u$, then both $Ω$ and $u$ are either rotationally symmetric or antipodally symmetric. Using the Alexandrov reflection method, we establish an analogous symmetry result for properly embedded constant mean curvature (CMC) surfaces with capillary boundaries that contain a simple curve of minimum distance to the origin (a neck). In the second part, we strengthen these conclusions for specific nonlinearities, including the eigenvalue problem ($f(x)=λx$), the Serrin problem ($f(x)=λx + c$), harmonic domains ($f(x)=c$), and nonlinearities of the form $f(x)=λx + x^β$, for constants $λ\geq 2$, $c \geq 0$, and $β\in (0,1)$. In these cases, we prove that the domain $Ω$ must be rotationally symmetric. Throughout the paper, we restrict our attention to the analytic setting in order to simplify the exposition and highlight the main ideas.

math.AP↗

arXiv2026

On the Dirichlet boundary value problem on Cartan-Hadamard manifolds

In this paper, we investigate the Dirichlet boundary value problem on Cartan-Hadamard manifolds, focusing on the non-existence of bounded (viscosity) solutions to semi-linear elliptic equations of the form $Δu + f(u) = 0$ in domains with prescribed asymptotic boundary, extending previous results by Bonorino and Klaser originally established for hyperbolic spaces. Using a novel comparison technique based on convex hypersurfaces inspired by Choi, Gálvez, and Lozano, we overcome the absence of totally geodesic foliations, which are instrumental in the hyperbolic space. Our results highlight the interplay between curvature, the spectrum of the Laplacian, and the geometry of the asymptotic boundary.

math.AP↗

arXiv2024

Rigidity results for Serrin's overdetermined problems in Riemannian manifolds

In this work, we are interested in studying Serrin's overdetermined problems in Riemannian manifolds. For manifolds endowed with a conformal vector field, we prove a Pohozoaev-type identity to show a Serrin's type rigidity result using the P-function approach introduced by Weinberger. We proceed with a conformal change to achieve this goal, starting from a geometric Pohozaev identity due to Schoen. Moreover, we obtain a symmetry result for the associated Dirichlet problem by using a generalized normalized wall shear stress bound.

math.DG↗

arXiv2023

An overdetermined eigenvalue problem and the Critical Catenoid conjecture

We consider the eigenvalue problem $Δ^{\mathbb{S}^2} ξ+ 2 ξ=0 $ in $ Ω$ and $ξ= 0 $ along $ \partial Ω$, being $Ω$ the complement of a disjoint and finite union of smooth and bounded simply connected regions in the two-sphere $\mathbb{S}^2$. Imposing that $|\nabla ξ|$ is locally constant along $\partial Ω$ and that $ξ$ has infinitely many maximum points, we are able to classify positive solutions as the rotationally symmetric ones. As a consequence, we obtain a characterization of the critical catenoid as the only embedded free boundary minimal annulus in the unit ball whose support function has infinitely many critical points.

math.AP↗