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Quasiconvex functions on regular trees

Leandro M. Del Pezzo, Nicolas Frevenza, Julio D. Rossi

Abstract

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.

Quasiconvex functions on regular trees

Abstract

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.

Paper Structure

This paper contains 12 sections, 11 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Settings, notations and statements
  3. Quasiconvexity for directed regular trees
  4. The quasiconvex envelope of a boundary datum
  5. The quasiconvex envelope of a function inside $\mathbb{T}_m$
  6. Proofs
  7. The quasiconvex envelope of a boundary datum
  8. The quasiconvex envelope of a function inside $\mathbb{T}_m$
  9. The quasiconvex envelope for monotone boundary data
  10. On the equations that characterize the convex and quasiconvex envelopes
  11. The Euclidean case
  12. The tree case

Key Result

Theorem 2.3

A function $u$ is $k-$quasiconvex if and only if

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Theorem 2.7
  • Corollary 2.8
  • Proposition 3.1
  • proof
  • ...and 12 more