The regularity of minima for the Dirichlet problem on BD. (English) Zbl 1442.49048
Many physically relevant variational problems describing the displacements of bodies subject to external forces are posed in the space BD of functions of bounded deformation. The space of BD displays the natural function space for a large class of variational integrals. In this paper, a regularity theory is developed from a Sobolev regularity and partial Hölder continuity perspective for minima of such functionals with linear growth, which essentially yields the same results known for the Dirichlet problems on BV of functions of bounded variations.
The generalised minima of the variational prolem are given as \[ \text{minimize } F[v]: = \int_\Omega f((\epsilon(v))dx, \quad \text{over } v \in{\mathcal D}_{u_0},\tag{a} \] where \({\mathcal D}_{u_0}\) is a suitable Dirichlet class and the integrand \(f\) is the convex function with linear growth. The main focus of the paper is on higher Sobolev and partial regularity for generalized minima of the variational principle (a). The corresponding results rely on the degenerate elliptic behavior of the integrands \(f\) known for the Dirichlet problem on BV. In the paper, new criteria are developed for the full gradients of generalized minima to exist as locally finite Radon measures.
While the paper is lengthy, it is clearly written, and the motivation behind the work is well explained and illustrated.
The generalised minima of the variational prolem are given as \[ \text{minimize } F[v]: = \int_\Omega f((\epsilon(v))dx, \quad \text{over } v \in{\mathcal D}_{u_0},\tag{a} \] where \({\mathcal D}_{u_0}\) is a suitable Dirichlet class and the integrand \(f\) is the convex function with linear growth. The main focus of the paper is on higher Sobolev and partial regularity for generalized minima of the variational principle (a). The corresponding results rely on the degenerate elliptic behavior of the integrands \(f\) known for the Dirichlet problem on BV. In the paper, new criteria are developed for the full gradients of generalized minima to exist as locally finite Radon measures.
While the paper is lengthy, it is clearly written, and the motivation behind the work is well explained and illustrated.
Reviewer: Bülent Karasözen (Ankara)
Keywords:
BD (functions of bounded deformation); Soboloev regularity; partial Hölder contimuity; Dirichlet problemReferences:
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