Gradient integrability for bounded BD-minimizers

2024/12/20 by Beck, Lisa, Eitler, Ferdinand, Gmeineder, Franz
#35B65 #35J60 #35J93 #49J45 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2412.16131

Abstract

We establish that locally bounded relaxed minimizers of degenerate elliptic symmetric gradient functionals on BD(Ω) have weak gradients in L_loc¹(Ω;ℝ^{n× n}). This is achieved for the sharp ellipticity range that is presently known to yield W_loc^1,1-regularity in the full gradient case on BV(Ω;ℝⁿ). As a consequence, we also obtain the first Sobolev regularity results for minimizers of the area-type functional on BD(Ω).

Related

• On the singular set of BV minimizers for non-autonomous functionals
• On BV^\mathbbA-Minimisers in two Dimensions
• Regularity for the Dirichlet problem on BD
• Gradient regularity for a class of elliptic obstacle problems
• Partial Regularity for Symmetric-Convex Functionals of Linear Growth