arXiv subjects

Marcelo Souza

Publications and source records attributed to Marcelo Souza.

3 recordsLinked to original sources

arXiv2024

Geometry of a Navigation problem: The $\lambda-$Funk Finsler Metrics

We investigate the travel time in a navigation problem from a geometric perspective. The setting involves an open subset of the Euclidean plane, representing a lake perturbed by a symmetric wind flow proportional to the distance from the origin. The Randers metric derived from this physical problem generalizes the well-known Euclidean metric on the Cartesian plane and the Funk metric on the unit disk. We obtain formulas for distances, or travel times, from point to point, from point to line, and vice-versa

math.DG↗

arXiv2016

On the study of a class of non-linear differential equations on compact Riemannian Manifolds

We study the existence of solutions of the non-linear differential equations on the compact Riemannian manifolds $(M^n,g), n\geq 2$, Δ_p u + a(x)u^{p-1} = λf(u,x), (E2) where $Δ_p$ is the $p-$laplacian, with $1<p<n$. The equation (E2) generalizes a equation considered by Aubin, where he has considered, a compact Riemannian manifold $(M,g)$, the differential equation ($p=2$) Δu + a(x)u = λf(u,x), (E1) where $a(x)$ is a $C^{\infty}$ function defined on $M$ and $f(u,x)$ is a $C^{\infty}$ function defined on $\mathbb{R}\times M$. We show that the equation (E2) has solution $(λ,u)$, where $λ\in \mathbb{R}$, $u \geq 0$, $u \not\equiv 0$ is a function $C^{1,α}$, $0 < α< 1$, if $f \in C^{\infty}$ satisfies some growth and parity conditions.

math.DG↗

arXiv2016

On the study of solutions for a non linear differential equation on compact Riemannian Manifolds

In this paper we study the existence of solutions for a class of non-linear differential equation on compact Riemannian manifolds. We establish a lower and upper solutions' method to show the existence of a smooth positive solution for the equation (EQ1) \begin{equation} \label{E4} Δu \ + \ a(x)u \ = \ f(x)F(u) \ + \ h(x)H(u), (EQ1) \end{equation} where \ $a, \ f, \ h$ \ are positive smooth functions on $M^n$, a $n-$dimensional compact Riemannian manifold, and \ $ F, \ H$ \ are non-decreasing smooth functions on $\mathbb{R}$. In \cite{djadli} the equation (EQ1) was studied when $F(u)=u^{2^{\ast}-1} $ and $H(u)=u^q$ in the Riemannian context, i.e., \begin{equation} \label{E3} Δu \ + \ a(x)u \ = \ f(x)u^{2^{\ast}-1} \ + \ h(x)u^q, (EQ2) \end{equation} \nd where \ $0 \ < \ q \ < 1$. In \cite{correa} Corrêa, Gonçalves and Melo studied an equation of the type equation (EQ2), in the Euclidean context.

math.DG↗