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On the attainment of boundary data in variational problems with linear growth

David Meyer

TL;DR

This work analyzes boundary attainment for convex variational problems with linear growth under Dirichlet-type conditions. It develops a BV-relaxation framework using the recession function $f^\infty$ and Anzellotti pairing, and introduces a domain-transformation strategy that reduces the problem to a ball with spherical-cap estimates to obtain trace control. The main contributions are two theorems (T1 for the scalar case and T2 for the vectorial case) showing that minimizers of the relaxed problem attain the prescribed boundary data in trace sense under mean-convex boundary conditions and suitable regularity of the boundary data (e.g., $u_0\in BV$ or $W^{\alpha,p}$ with $\alpha p\ge 2$); these results extend to systems under a quasi-isotropy assumption on the integrand and have applications to ROF-type functionals and trace questions for least-gradient problems. The paper also establishes uniqueness results in various settings, discusses optimality of assumptions, provides a corollary removing the $L^2$ requirement in the scalar case, and constructs smooth counterexamples when quasi-isotropy fails, highlighting the sharpness and limitations of the theory.

Abstract

It is well-known that convex variational problems with linear growth and Dirichlet boundary conditions might not have minimizers if the boundary condition is not suitably relaxed. We show that for a wide range of integrands, including the least gradient problem and the non-parametric Plateau problem, and under suitable mean-convexity conditions of the boundary, minimizers of the relaxed problem attain the boundary data in the trace sense if it lies in $BV$ or $W^{α,p}$ with $αp\geq 2$ without any kind of continuity assumption. Unlike previous works, our methods are also able to treat systems under a certain quasi-isotropy assumption on the integrand. We further show that without this quasi-isotropy assumption, smooth counterexamples on uniformly convex domains exist. Further applications to the uniqueness of minimizers and to open problems about the ROF functional with Dirichlet boundary conditions, and to the trace space of functions of least gradient are given.

On the attainment of boundary data in variational problems with linear growth

TL;DR

This work analyzes boundary attainment for convex variational problems with linear growth under Dirichlet-type conditions. It develops a BV-relaxation framework using the recession function and Anzellotti pairing, and introduces a domain-transformation strategy that reduces the problem to a ball with spherical-cap estimates to obtain trace control. The main contributions are two theorems (T1 for the scalar case and T2 for the vectorial case) showing that minimizers of the relaxed problem attain the prescribed boundary data in trace sense under mean-convex boundary conditions and suitable regularity of the boundary data (e.g., or with ); these results extend to systems under a quasi-isotropy assumption on the integrand and have applications to ROF-type functionals and trace questions for least-gradient problems. The paper also establishes uniqueness results in various settings, discusses optimality of assumptions, provides a corollary removing the requirement in the scalar case, and constructs smooth counterexamples when quasi-isotropy fails, highlighting the sharpness and limitations of the theory.

Abstract

It is well-known that convex variational problems with linear growth and Dirichlet boundary conditions might not have minimizers if the boundary condition is not suitably relaxed. We show that for a wide range of integrands, including the least gradient problem and the non-parametric Plateau problem, and under suitable mean-convexity conditions of the boundary, minimizers of the relaxed problem attain the boundary data in the trace sense if it lies in or with without any kind of continuity assumption. Unlike previous works, our methods are also able to treat systems under a certain quasi-isotropy assumption on the integrand. We further show that without this quasi-isotropy assumption, smooth counterexamples on uniformly convex domains exist. Further applications to the uniqueness of minimizers and to open problems about the ROF functional with Dirichlet boundary conditions, and to the trace space of functions of least gradient are given.

Paper Structure

This paper contains 40 sections, 29 theorems, 307 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Precise Set-up and main results
  3. The scalar case $n=1$
  4. The vectorial case $n\geq 1$
  5. Further discussion
  6. Uniqueness
  7. The Dirichlet problem for the Rudin-Osher-Fatemi functional
  8. The trace space of functions of least gradient
  9. Comments on the optimality of the assumptions
  10. Organisation of the paper
  11. Preliminaries and notation
  12. Notational conventions
  13. The space BV
  14. The Anzellotti Pairing
  15. Elements of convex analysis
  16. ...and 25 more sections

Key Result

Theorem 1.1

Suppose that the assumptions A1, A2, A3, A4, A5 and A6 above hold and that $u\in BV(\Omega)\cap L^2(\Omega)$ is a minimizer of the problem gen rel prob. Then in the trace sense.

Figures (2)

  • Figure 1: Sketch of the geometry in the proof of Theorem \ref{['T1']}
  • Figure 2: Sketch of the geometry in the proof of Theorem \ref{['T2']}

Theorems & Definitions (47)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7
  • proof
  • Corollary 1.8
  • ...and 37 more