The Regularity of Minima for the Dirichlet Problem on BD

Abstract

AbstractWe establish that the Dirichlet problem for linear growth functionals on $${\text {BD}}$$ BD , the functions of bounded deformation, gives rise to the same unconditional Sobolev and partial $${\text {C}}^{1,\alpha }$$ C 1 , α -regularity theory as presently available for the full gradient Dirichlet problem on $${\text {BV}}$$ BV . Functions of bounded deformation play an important role in, for example plasticity, however, by Ornstein’s non-inequality, contain $${\text {BV}}$$ BV as a proper subspace. Thus, techniques to establish regularity by full gradient methods for variational problems on BV do not apply here. In particular, applying to all generalised minima (that is, minima of a suitably relaxed problem) despite their non-uniqueness and reaching the ellipticity ranges known from the $${\text {BV}}$$ BV -case, this paper extends previous Sobolev regularity results by Gmeineder and Kristensen (in J Calc Var 58:56, 2019) in an optimal way.