We 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.

中文翻译：

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BD 上狄利克雷问题的最小值的规律

我们确定 $${\text {BD}}$$ BD 上线性增长函数的 Dirichlet 问题，即有界变形函数，产生相同的无条件 Sobolev 和部分 $${\text {C}}^{ 1,\alpha }$$ C 1 , α -正则理论目前可用于 $${\text {BV}}$$ BV 上的全梯度狄利克雷问题。有界变形函数在例如塑性方面发挥重要作用，然而，根据 Ornstein 的非不等式，包含 $${\text {BV}}$$ BV 作为适当的子空间。因此，通过全梯度方法为 BV 上的变分问题建立规律性的技术不适用于这里。特别是，适用于所有广义最小值（即适当松弛问题的最小值），尽管它们是非唯一性的，并且达到了从 $${\text {BV}}$$ BV -case 已知的椭圆度范围，