arXiv subjects

Renan J. S. Isneri

Publications and source records attributed to Renan J. S. Isneri.

7 recordsLinked to original sources

arXiv2026

Prescribed--Energy Connecting Orbits for Quasilinear Conservative Systems

We consider quasilinear conservative systems \[ (ϕ(|\dot q|)\dot q)'=\nabla V(q), \qquad q\in\R^{N}, \] with $Φ$-growth kinetic term and potential $V\in C^{1}(\R^{N};\R)$. Assuming that for some $c\in\R$ the sublevel set $\{V\le c\}$ splits into two disjoint closed subsets $\mathcal V_c^{-}$ and $\mathcal V_c^{+}$, we prove the existence of trajectories $q_c$ with prescribed energy $-c$ connecting $\mathcal V_c^{-}$ and $\mathcal V_c^{+}$, obtained through an energy-constrained variational method. Although the construction yields weak solutions in an Orlicz-Sobolev setting, minimal $c$-connections are shown to be classical $C^2$ trajectories satisfying the strong energy identity $E_{q_c}\equiv -c$. The resulting entire trajectories include heteroclinic, homoclinic, and brake-type orbits. Applications to double-well, Duffing-type, and multiple pendulum systems are discussed.

math.AP↗

arXiv2026

On the origin of the Gibbons conjecture

The term Gibbons conjecture is widely used in connection with symmetry results for the Allen-Cahn equation. However, its origin is less transparent than its frequent citation suggests. In this note, we revisit its emergence, tracing it to a 1995 paper by Carbou and to subsequent developments in the literature. We argue that the attribution likely arose from informal communication rather than from a formally stated conjecture, illustrating how mathematical terminology may develop through transmission and collective usage.

math.HO↗

arXiv2025

Asymptotic behavior for a class of damped second-order gradient systems via Lyapunov method

In this work we study the asymptotic behavior of a class of damped second-order gradient systems $$ \ddot{u}(t) + a\dot{u}(t) + \nabla W(u(t)) = 0, $$ under assumptions ensuring local convexity of the potential near equilibrium and coercivity at infinity. By introducing a Lyapunov functional adapted to the geometry of the system, we establish uniform asymptotic stability of the equilibrium for all $a \in (0,a_0]$, together with exponential decay when the potential satisfies a quadratic control near its minimum. Furthermore, complementary arguments based on semigroup theory reveal the existence of a global attractor. We also present numerical simulations for some $W$ potentials that illustrate the behavior of trajectories near equilibrium, in both dissipative and conservative regimes.

math.CA↗

arXiv2025

Norms and Non-Equivalence in Infinite-Dimensional Banach Spaces

This work explores the interaction between different norms in infinite-dimensional vector spaces, focusing on their impact on Banach space structures and topological properties. We examine norms induced by bijective linear maps, the existence of non-equivalent norms in Banach spaces, and the role of Hamel bases in normed spaces.

math.FA↗

arXiv2024

Heteroclinic solutions for some classes of prescribed mean curvature equations in whole $\mathbb{R}^2$

The purpose of this paper consists in using variational methods to establish the existence of heteroclinic solutions for some classes of prescribed mean curvature equations of the type $$ -div\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) + A(εx,y)V'(u)=0~~\text{ in }~~\mathbb{R}^2, $$ where $ε>0$ and $V$ is a double-well potential with minima at $t=α$ and $t=β$ with $α<β$. Here, we consider some class of functions $A(x,y)$ that are oscillatory in the variable $y$ and satisfy different geometric conditions such as periodicity in all variables or asymptotically periodic at infinity.

math.AP↗

arXiv2024

Uniqueness of Heteroclinic Solutions in a Class of Autonomous Quasilinear ODE Problems

In this paper, we prove the existence, uniqueness and qualitative properties of heteroclinic solution for a class of autonomous quasilinear ordinary differential equations of the Allen-Cahn type given by $$ -\left(ϕ(|u'|)u'\right)'+V'(u)=0~~\text{ in }~~\mathbb{R}, $$ where $V$ is a double-well potential with minima at $t=\pmα$ and $ϕ:(0,+\infty)\to(0,+\infty)$ is a $C^1$ function satisfying some technical assumptions. Our results include the classic case $ϕ(t)=t^{p-2}$, which is related to the celebrated $p$-Laplacian operator, presenting the explicit solution in this specific scenario. Moreover, we also study the case $ϕ(t)=\frac{1}{\sqrt{1+t^2}}$, which is directly associated with the prescribed mean curvature operator.

math.AP↗

arXiv2024

Saddle solutions for Allen-Cahn type equations involving the prescribed mean curvature operator

The goal of this paper is to investigate the existence of saddle solutions for some classes of elliptic partial differential equations of the Allen-Cahn type, formulated as follows: \begin{equation*} -div\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) + A(x,y)V'(u)=0~~\text{ in }~~\mathbb{R}^2. \end{equation*} Here, the function $A:\mathbb{R}^2\to\mathbb{R}$ exhibits periodicity in all its arguments, while $V:\mathbb{R}\to\mathbb{R}$ characterizes a double-well symmetric potential with minima at $t=\pmα$.

math.AP↗