Abstract
We establish that the Dirichlet problem for linear growth functionals on {text {BD}}, the functions of bounded deformation, gives rise to the same unconditional Sobolev and partial {text {C}}^{1,alpha }-regularity theory as presently available for the full gradient Dirichlet problem on {text {BV}}. Functions of bounded deformation play an important role in, for example plasticity, however, by Ornstein’s non-inequality, contain {text {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}}-case, this paper extends previous Sobolev regularity results by Gmeineder and Kristensen (in J Calc Var 58:56, 2019) in an optimal way.
Highlights
We establish that the Dirichlet problem for linear growth functionals on BD, the functions of bounded deformation, gives rise to the same unconditional Sobolev and partial C1,α-regularity theory as presently available for the full gradient Dirichlet problem on BV
By the coerciveness considerations outlined below, this space displays the natural function space setup for a vast class of variational integrals. For minima of such functionals, the present paper aims to develop a regularity theory which—from a Sobolev regularity and partial Hölder continuity perspective—essentially yields the same results which are presently known for the Dirichlet problem on BV
When linear growth functionals are considered—that is, c1|z|−γ g(z) c2(1 + |z|) for some c1, c2, γ > 0 and all z ∈ RN×n— compactness considerations lead to the study of minima of a suitably relaxed problem on BV, cf
Summary
Toward a unifying regularity theory for the Dirichlet problem on BD, we begin by giving the underlying setup first. One usually imposes additional hypotheses—such as local boundedness, cf [22]—on minima in order to obtain regularity results, and such conditions can be justified for a variety of situations, so for instance by maximum principles or, in the radial situation, Moser-type L∞-bounds This distinction of ellipticity regimes enters in the BV-theory for full gradient functionals. The comparison argument forces us to control V -function type distances from a given generalised minimiser to its mollifications by the symmetric gradients only While this is a consequence of the fundamental theorem of calculus in the BV-context, the requisite estimates must be accessed without appealing to the full gradients.
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