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Young functions on varifolds. Part I. Functional analytic foundations

Hsin-Chuang Chou

TL;DR

This paper develops a functional-analytic foundation for Young functions on varifolds, introducing graph measures to model the convergence of pairs consisting of surfaces and multi-valued functions. It defines and analyzes the convergence via graph measures, proves a compactness theorem for pairs of rectifiable varifolds and Young functions, and establishes a disintegration theorem to underpin this compactness. The work also introduces robust spaces of probability Radon measures with operations on Young functions, together with carefully designed test-function spaces E(Y) and H(U×Y, R^n) that support a differentiability framework and embedding results. Collectively, these results set the stage for future development of differentiability for Young functions on varifolds and for integrating with Almgren's Q-valued function theory. The approach provides a rigorous bridge between geometric measure theory and functional-analytic methods for multi-valued analysis with potential applications to variational problems in higher codimension.

Abstract

In this paper, we study Young functions, a measure-theoretic model for multiple-valued functions, and the convergence of pairs of measures and Young functions via their associated graph measures. This setting allows us to study the convergence of pairs of surfaces and functions thereon, and a compactness theorem is immediate. To develop notions of differentiability of Young functions in the upcoming papers, we also introduce and investigate several test function spaces.

Young functions on varifolds. Part I. Functional analytic foundations

TL;DR

This paper develops a functional-analytic foundation for Young functions on varifolds, introducing graph measures to model the convergence of pairs consisting of surfaces and multi-valued functions. It defines and analyzes the convergence via graph measures, proves a compactness theorem for pairs of rectifiable varifolds and Young functions, and establishes a disintegration theorem to underpin this compactness. The work also introduces robust spaces of probability Radon measures with operations on Young functions, together with carefully designed test-function spaces E(Y) and H(U×Y, R^n) that support a differentiability framework and embedding results. Collectively, these results set the stage for future development of differentiability for Young functions on varifolds and for integrating with Almgren's Q-valued function theory. The approach provides a rigorous bridge between geometric measure theory and functional-analytic methods for multi-valued analysis with potential applications to variational problems in higher codimension.

Abstract

In this paper, we study Young functions, a measure-theoretic model for multiple-valued functions, and the convergence of pairs of measures and Young functions via their associated graph measures. This setting allows us to study the convergence of pairs of surfaces and functions thereon, and a compactness theorem is immediate. To develop notions of differentiability of Young functions in the upcoming papers, we also introduce and investigate several test function spaces.

Paper Structure

This paper contains 5 sections, 42 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Inductive limit and topological tensor product
  3. Young functions and graph measures
  4. The space of probability Radon measures and operations on Young functions
  5. The test function spaces

Key Result

Theorem 1

Suppose $k$, $m$, and $n$ are positive integers, $m \leq n$, $V_i$ is a sequence of $m$-dimensional varifolds in $\mathbf{R}^n$ that converges to $V$, and $f_i$ is a sequence of $\|V_i\|$ Young functions of type $\mathbf{R}^k$ such that whenever $K$ is a compact subset of $\mathbf{R}^n$. Then, there exists a $\|V\|$ Young function $f$ of type $\mathbf{R}^k$ such that, possibly passing to a subseq

Theorems & Definitions (147)

  • Definition : see \ref{['definition: Young function']}
  • Definition : see \ref{['definition: graph measure']}
  • Theorem : see \ref{['theorem: compactness theorem for pairs of rectifiable varifolds and Young functions']}
  • Definition 2.1: see Kelley75_MR0370454
  • Definition 2.2: see Bourbaki_TheoryOfSet_MR2102219 and Bourbaki_TVS_MR910295
  • Remark 2.3: see Bourbaki_TheoryOfSet_MR2102219
  • Definition 2.4: see Bourbaki_TVS_MR910295
  • Remark 2.5: see Bourbaki_TVS_MR910295
  • Remark 2.6: see Bourbaki_TVS_MR910295
  • Remark 2.7: see Bourbaki_TheoryOfSet_MR2102219
  • ...and 137 more