The Regularity of Minima for the Dirichlet Problem on BD
Franz Gmeineder
Abstract
Abstract We establish that the Dirichlet problem for linear growth functionals on $${\text {BD}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>BD</mml:mtext></mml:math> , the functions of bounded deformation, gives rise to the same unconditional Sobolev and partial $${\text {C}}^{1,\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mtext>C</mml:mtext></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:msup></mml:math> -regularity theory as presently available for the full gradient Dirichlet problem on $${\text {BV}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>BV</mml:mtext></mml:math> . Functions of bounded deformation play an important role in, for example plasticity, however, by Ornstein ’s non-inequality, contain $${\text {BV}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>BV</mml:mtext></mml:math> 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 $${\text {BV}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>BV</mml:mtext></mml:math> -case, this paper extends previous Sobolev regularity results by Gmeineder and Kristensen (in J Calc Var 58:56, 2019) in an optimal way.