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The two membranes problem in a regular tree

Irene Gonzálvez, Alfredo Miranda, Julio D. Rossi

TL;DR

The paper addresses the two membranes problem for mean value operators on a regular tree, formulating the coupled obstacle system for two operators $L_1$ and $L_2$ and proving existence of a bounded solution under solvability conditions on the boundary data and source terms. It obtains the solution by iterating obstacle problems from above and below, yielding a limit pair $(u,v)$ that satisfies the coupled inequalities and boundary data, and shows the coincidence set is finite when the boundary data are strictly separated. It further provides a probabilistic interpretation via a finite-horizon two-player zero-sum game on two trees, with value functions converging to the tree solution as the horizon grows. Overall, the work extends the classical two membranes theory to discrete, nonlocal graph settings and connects obstacle problems with mean-value operators and game-theoretic interpretations on trees.

Abstract

In this paper we study the two membranes problem for operators given in terms of a mean value formula on a regular tree. We show existence of solutions under adequate conditions on the boundary data and the involved source terms. We also show that, when the boundary data are strictly separated, the coincidence set is separated from the boundary and thus it contains only a finite number of nodes.

The two membranes problem in a regular tree

TL;DR

The paper addresses the two membranes problem for mean value operators on a regular tree, formulating the coupled obstacle system for two operators and and proving existence of a bounded solution under solvability conditions on the boundary data and source terms. It obtains the solution by iterating obstacle problems from above and below, yielding a limit pair that satisfies the coupled inequalities and boundary data, and shows the coincidence set is finite when the boundary data are strictly separated. It further provides a probabilistic interpretation via a finite-horizon two-player zero-sum game on two trees, with value functions converging to the tree solution as the horizon grows. Overall, the work extends the classical two membranes theory to discrete, nonlocal graph settings and connects obstacle problems with mean-value operators and game-theoretic interpretations on trees.

Abstract

In this paper we study the two membranes problem for operators given in terms of a mean value formula on a regular tree. We show existence of solutions under adequate conditions on the boundary data and the involved source terms. We also show that, when the boundary data are strictly separated, the coincidence set is separated from the boundary and thus it contains only a finite number of nodes.
Paper Structure (6 sections, 6 theorems, 180 equations)

This paper contains 6 sections, 6 theorems, 180 equations.

Table of Contents

  1. Introduction.
  2. Conditions for existence of bounded solutions to a single equation
  3. The two membranes problem.
  4. The obstacle problem on trees.
  5. The two membranes problem. Existence via iterations of the obstacle problem.
  6. A probabilistic interpretation of the two membranes problem in the tree

Key Result

Theorem 1.1

Given two averaging operators, $L_{1}$ and $L_{2}$ defined as in L1 and L2 (that involve two different parameters $\beta_1$, $\beta_2$), two different functions $h_1,h_2: \mathbb{T} \rightarrow \mathbb{R}$ and two continuous boundary conditions $f,g:[0,1]\rightarrow\mathbb{R}$ with $f > g$ in $[0,1] then, there exists a pair $(u,v):\mathbb{T}\mapsto \mathbb{R}$ that is a solution to the two membra

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1.1
  • Theorem 1.2
  • proof : Proof of Theorem \ref{['teo.intro.una.eacuacion']}
  • Remark 1
  • Remark 2
  • Lemma 2.1
  • ...and 9 more