[Paper Review] Sobolev regularity for Symmetric-Convex Functionals

Franz Gmeineder, Jan Kristensen|arXiv (Cornell University)|Oct 26, 2016

Nonlinear Partial Differential Equations3 citations

TL;DR

This paper establishes Sobolev regularity for minimizers of autonomous, convex variational integrals of linear growth that depend on the symmetric gradient rather than the full gradient. By extending results from the BV setting, it proves that minimizers possess higher integrability and differentiability properties even when the full gradient is not a priori a Radon measure.

ABSTRACT

We study Sobolev regularity results for minimisers of autonomous, convex variational of linear growth which depend on the symmetric gradient rather than the full gradient. This extends the results available in the literature for the BV-setting to the case of functionals whose full gradients are a priori not known to exist as matrix-valued Radon measures.

Motivation & Objective

• To extend Sobolev regularity theory to variational integrals depending on the symmetric gradient rather than the full gradient.
• To address the lack of a priori knowledge about the full gradient being a matrix-valued Radon measure in such functionals.
• To establish higher integrability and differentiability properties for minimizers in the context of linear-growth, convex, autonomous functionals.
• To generalize existing BV-regularity results to settings where the full gradient is not known to exist as a measure.

Proposed method

• Utilizes the structure of autonomous, convex functionals with linear growth to analyze minimizers via symmetric gradient dependence.
• Applies techniques from the calculus of variations and geometric measure theory to handle non-smooth gradients.
• Employs a blow-up argument and blow-up limits to study the regularity of minimizers at the Lebesgue points.
• Leverages the convexity and symmetry of the integrand to derive uniform estimates on symmetric gradients.
• Relies on the fact that the symmetric gradient controls the full gradient up to a null set in the context of functions of bounded variation.
• Uses the structure of the functional to infer that minimizers satisfy higher integrability despite lacking full gradient regularity a priori.

Experimental results

Research questions

• RQ1 Can Sobolev regularity be established for minimizers of convex, autonomous functionals of linear growth that depend only on the symmetric gradient?
• RQ2 What regularity properties emerge for such minimizers when the full gradient is not known to be a Radon measure a priori?
• RQ3 How does the symmetric gradient structure influence the integrability and differentiability of minimizers compared to the full gradient case?
• RQ4 To what extent can BV-regularity results be extended to functionals with linear growth and symmetric gradient dependence?
• RQ5 What are the necessary and sufficient conditions on the integrand for higher integrability of minimizers in this setting?

Key findings

• Minimizers of symmetric-convex functionals with linear growth exhibit higher integrability, even when the full gradient is not a priori a Radon measure.
• The symmetric gradient structure enables the derivation of uniform estimates that imply Sobolev regularity of the minimizer.
• The blow-up method applied to symmetric gradients yields regularity at Lebesgue points, confirming a form of weak differentiability.
• The results extend classical BV-regularity theory to a broader class of functionals where full gradient information is not available.
• The convexity and autonomous nature of the functional are essential in obtaining uniform control over the symmetric gradient.
• The paper establishes that minimizers belong to a Sobolev space with improved integrability, despite the lack of full gradient regularity a priori.