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The local Burkholder functional, quasiconvexity and Geometric Function Theory

Kari Astala, Daniel Faraco, André Guerra, Aleksis Koski, Jan Kristensen

TL;DR

This work advances the understanding of quasiconvexity for Burkholder-type functionals in the planar setting, establishing closed $W^{1,p}$-quasiconvexity for the local Burkholder functional when tested on $K$-quasiconformal cones and for a range of exponents $p>\tfrac{2K}{K+1}$. It introduces the sharp Burkholder area inequality for principal maps, analyzes the $p\to2$ limit yielding the rank-one convex but non-polyconvex functional ${\mathscr F}$ and its Shield transform ${\mathscr W}$, and proves quasiconvexity of ${\mathscr W}$ as well as related higher-integrability results. By combining gradient Young measures with the Direct Method, the paper derives lower semicontinuity and existence of minimizers for Burkholder energies and, more broadly, for functionals with additive volumetric-isochoric splits, including non-polyconvex elastic energies. The Shield transformation provides a mechanism to generate new quasiconvex functionals, while the additive-split analysis connects these variational questions to Morrey-type problems in Geometric Function Theory. Overall, the results yield new existence and regularity insights for nonlinear elasticity models and deepen the connection between quasiconvexity, gradient Young measures, and planar quasiconformal theory, with sharp area-type inequalities and higher integrability consequences.

Abstract

We show that the local Burkholder functional $\mathcal B_K$ is quasiconvex. In the limit of $p$ going to 2 we find a class of non-polyconvex functionals which are quasiconvex on the set of matrices with positive determinant. In order to prove the validity of lower semicontinuity arguments in this setting, we show that the Burkholder functionals satisfy a sharp extension of the classical function theoretic area formula. As a corollary, in addition to functionals in geometric function theory, one finds new classes of non-polyconvex functionals, degenerating as the determinant vanishes, for which there is existence of minimizers.

The local Burkholder functional, quasiconvexity and Geometric Function Theory

TL;DR

This work advances the understanding of quasiconvexity for Burkholder-type functionals in the planar setting, establishing closed -quasiconvexity for the local Burkholder functional when tested on -quasiconformal cones and for a range of exponents . It introduces the sharp Burkholder area inequality for principal maps, analyzes the limit yielding the rank-one convex but non-polyconvex functional and its Shield transform , and proves quasiconvexity of as well as related higher-integrability results. By combining gradient Young measures with the Direct Method, the paper derives lower semicontinuity and existence of minimizers for Burkholder energies and, more broadly, for functionals with additive volumetric-isochoric splits, including non-polyconvex elastic energies. The Shield transformation provides a mechanism to generate new quasiconvex functionals, while the additive-split analysis connects these variational questions to Morrey-type problems in Geometric Function Theory. Overall, the results yield new existence and regularity insights for nonlinear elasticity models and deepen the connection between quasiconvexity, gradient Young measures, and planar quasiconformal theory, with sharp area-type inequalities and higher integrability consequences.

Abstract

We show that the local Burkholder functional is quasiconvex. In the limit of going to 2 we find a class of non-polyconvex functionals which are quasiconvex on the set of matrices with positive determinant. In order to prove the validity of lower semicontinuity arguments in this setting, we show that the Burkholder functionals satisfy a sharp extension of the classical function theoretic area formula. As a corollary, in addition to functionals in geometric function theory, one finds new classes of non-polyconvex functionals, degenerating as the determinant vanishes, for which there is existence of minimizers.
Paper Structure (26 sections, 49 theorems, 293 equations)

This paper contains 26 sections, 49 theorems, 293 equations.

Table of Contents

  1. Introduction
  2. The Burkholder functional
  3. The Burkholder area inequality
  4. Functionals for Nonlinear Elasticity, as $p \to 2$.
  5. Lower semicontinuity and existence of minimizers
  6. Extended real-valued functionals and Young measures
  7. Quasiconvexity and rank-one convexity
  8. Gradient Young measures
  9. Closed quasiconvexity and closed rank-one convexity
  10. Rank-one convexity and radial maps
  11. Principal mappings
  12. Quasiregular gradient Young measures
  13. The Burkholder functionals
  14. Weighted integral estimates and the Burkholder functionals
  15. Burkholder functional and radial mappings
  16. ...and 11 more sections

Key Result

Theorem 1.3

Let $p\geq 2$. For any $A \in \mathbb{R}^{2 \times 2}$ and any $f \in A+ \mathrm{W}^{1,2}_0(\Omega , \mathbb{R}^2)$ such that ${\bf B}_p(\textup{D} f) \leq 0$ a.e. in $\Omega$, we have

Theorems & Definitions (114)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6: Burkholder Area Inequality
  • Corollary 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 104 more