arXiv subjects

Giovanni da Silva Nunes

Publications and source records attributed to Giovanni da Silva Nunes.

2 recordsLinked to original sources

arXiv2026

An alternative solvability criterion for the Dirichlet problem for the minimal surface equation and an application to the mean curvature flow

We propose an alternative condition for the solvability of the Dirichlet problem for the minimal surface equation that applies to non-mean convex domains. We introduce a structural condition, obtained from a second-order ordinary differential equation, which allows the construction of explicit boundary barriers and it can also be applied to unbounded domains. In the setting of Hadamard manifolds, this condition relates the geometry of the domain to the admissible boundary data in a direct way. In Euclidean space, the condition leads to solvability under geometric hypotheses of a different nature from those in the classical Jenkins-Serrin theory, and in some configurations it applies where the Jenkins-Serrin method does not. A central point of the present approach is that the geometric restrictions and the boundary data enter independently. The same barrier construction can be used for graphical mean curvature flow. This yields short-time existence with prescribed boundary values even when the boundary of the domain is not mean convex. When mean convexity is present, one recovers the classical graphical setting.

math.AP↗

arXiv2025

A lower bound of the energy functional of a class of vector fields and a characterization of the sphere

Let $M$ be a compact, orientable, $n$-dimensional Riemannian manifold, $n\geq2$, and let $F$ be the energy functional acting on the space $Ξ(M)$ of $C^{\infty }$ vector fields of $M$, \[ F(X):=\frac{\int_{M}\left\Vert \nabla X\right\Vert ^{2}dM}{\int_{M}\left\Vert X\right\Vert ^{2}dM}, X\in Ξ(M)\backslash\{0\}. % \] Let $G\in\operatorname*{Iso}\left(M\right)$ be a compact Lie subgroup of the isometry group of $M$ acting with cohomogeneity $1$ on $M$. Assume that any isotropy subgroup of $G$ is non trivial and acts with no fixed points on the tangent spaces of $M$, except at the null vectors. We prove in this note that under these hypothesis, if the Ricci curvature $\operatorname*{Ric}\nolimits_{M}$ of $M$ has the lower bound $\operatorname*{Ric}\nolimits_{M}\geq(n-1)k^{2}$, then $\operatorname*{F(X)}\geq(n-1)k^{2}$, for any $G$-invariant vector field $X\in Ξ(M)\backslash\{0\}$, and the equality occurs if and only if $M$ is isometric to the n-dimensional sphere $\mathbb{S}^{n}_k$ of constant sectional curvature $k^{2}$. In this case $X$ is an infimum of $F$ on $Ξ(\mathbb{S}^{n}_k).$

math.DG↗