arXiv subjects

Van Phu Cuong Le

Publications and source records attributed to Van Phu Cuong Le.

6 recordsLinked to original sources

arXiv2026

On fractional semilinear wave equations in non-cylindrical domains

In this paper, we investigate a class of semilinear wave equations in non-cylindrical time-dependent domains, subject to exterior homogeneous Dirichlet conditions. Under mild regularity and monotonicity assumptions on the evolving spatial domains, we establish existence of weak solutions by two different methods: a constructive time-discretization scheme and a penalty approach. The analysis applies to nonlocal fractional Laplacians and potentials with Lipschitz continuous gradient, and to vector-valued maps.

math.AP↗

arXiv2025

$Γ$-convergence of the $p$-Dirichlet energy for manifold-valued maps

We prove a $Γ$-convergence result for the $p$-Dirichlet energy functional defined on maps from a smooth bounded domain $Ω\subseteq \mathbb{R}^{n+k}$ to $\mathscr{N}$, a $(k-2)$-connected and smooth closed Riemannian manifold with Abelian fundamental group, where $n$ and $k$ are integers, $n \geq 0$, $k \geq 2$. We focus on the regime $p \to~k^-$ under Dirichlet boundary conditions. The result provides a description of the asymptotic behavior of the $\textit{topological singular sets}$ for families of $\mathscr{N}$-valued Sobolev maps which satisfy suitable energy bounds. Such topological singular sets are $n$-dimensional flat chains with coefficients in $π_{k-1}(\mathscr{N})$ endowed with a suitable norm. As a consequence of our main result, it follows that the topological singular sets of energy minimizing $p$-harmonic maps converge to a $n$-dimensional flat chain $S$ with coefficients in $π_{k-1}(\mathscr{N})$ which has finite mass and solves the Plateau problem within the homology class associated to the boundary datum.

math.AP↗

arXiv2024

Energy-minimizing torus-valued maps with prescribed singularities, Plateau's problem, and BV-lifting

In this paper, we investigate the relation between energy-minimizing torus-valued maps with prescribed singularities, the lifting problem for torus-valued maps in the space BV, and Plateau's problem for vectorial currents, in codimension one. First, we show that the infimum of the $W^{1,1}$-seminorm among all maps with values in the $k$-dimensional flat torus and prescribed topological singularities $S$ is equal to the minimum of the mass among all $\textit{normal}$ $\mathbb{R}^k$-currents, of codimension one, bounded by $S$. Then, we show that the minimum of the $BV$-energy among all liftings of a given torus-valued $W^{1,1}$-map $\textbf{u}$ can be expressed in terms of the minimum mass among all $\textit{integral}$ $\mathbb{Z}^k$-currents, of codimension one, bounded by the singularities of $\textbf{u}$. As a byproduct of our analysis, we provide a bound for the solution of the integral Plateau problem, in codimension one, in terms of Plateau's problem for normal currents.

math.OC↗

arXiv2023

Energy minimizing maps with prescribed singularities and Gilbert-Steiner optimal networks

We investigate the relation between energy minimizing maps valued into spheres having topological singularities at given points and optimal networks connecting them (e.g. Steiner trees, Gilbert-Steiner irrigation networks). We show the equivalence of the corresponding variational problems, interpreting in particular the branched optimal transport problem as a homological Plateau problem for rectifiable currents with values in a suitable normed group. This generalizes the pioneering work by Brezis, Coron and Lieb [10].

math.OC↗

arXiv2022

Weak solutions for nonlinear waves in adhesive phenomena

We discuss a notion of weak solution for a semilinear wave equation that models the interaction of an elastic body with a rigid substrate through an adhesive layer, relying on results in [2]. Our analysis embraces the vector-valued case in arbitrary dimension as well as the case of non-local operators (e.g. fractional Laplacian).

math.AP↗

arXiv2021

On the obstacle problem for fractional semilinear wave equations

We prove existence of weak solutions to the obstacle problem for semilinear wave equations (including the fractional case) by using a suitable approximating scheme in the spirit of minimizing movements. This extends the results in [9], where the linear case was treated. In addition, we deduce some compactness properties of concentration sets (e.g. moving interfaces) when dealing with singular limits of certain nonlinear wave equations.

math.AP↗