arXiv subjects

Nicolas Frevenza

Publications and source records attributed to Nicolas Frevenza.

3 recordsLinked to original sources

arXiv2025

The evolution equation and the eigenvalue problem for the Laplacian in a regular tree

In this paper, our main goal is to study the evolution problem associated with the Laplacian operator with Dirichlet boundary conditions on a regular tree. To this end, we place special emphasis on the associated first eigenvalue problem, which provides the fundamental tool for describing the long-time dynamics. First, we prove existence and uniqueness of solutions when the initial condition is compatible with the boundary condition. Next, we address the asymptotic behavior of the solutions and show that they decay to zero exponentially fast. This decay rate is determined by the associated first eigenvalue, which we also analyze in detail.

math.AP↗

arXiv2020

Quasiconvex functions on regular trees

We introduce a definition of a quasiconvex function on an infinite directed regular tree that depends on what we understood by a segment on the tree. Our definition is based on thinking on segments as sub-trees with the root as the midpoint of the segment. A convex set in the tree is then a subset such that it contains every midpoint of every segment with terminal nodes in the set. Then a quasiconvex function is a real map on the tree such that every level set is a convex set. For this concept of quasiconvex functions on a tree, we show that given a continuous boundary datum there exists a unique quasiconvex envelope on the tree and we characterize the equation that this envelope satisfies. It turns out that this equation is a mean value property that involves a median among values of the function on successors of a given vertex. We also relate the quasiconvex envelope of a function defined inside the tree with the solution of an obstacle problem for this characteristic equation.

math.AP↗

arXiv2019

Convex envelopes on Trees

We introduce two notions of convexity for an infinite regular tree. For these two notions we show that given a continuous boundary datum there exists a unique convex envelope on the tree and characterize the equation that this envelope satisfies. We also relate the equation with two versions of the Laplacian on the tree. Moreover, for a function defined on the tree, the convex envelope turns out to be the solution to the obstacle problem for this equation.

math.AP↗