A Simple Partial Regularity Proof for Minimizers of Variational Integrals

Abstract

Abstract. We consider multi-dimensional variational integrals

Sobolev regularity for convex functionals on BD

“…However, note that if f : Ω × R n × R n×n sym → R satisfies a splitting condition f (x, y, z) = f 1 (x, z) + f 2 (x, y) for some strongly symmetric quasiconvex integrand f 1 : Ω × R n×n sym → R of linear growth and f 2 : Ω × R n → R being convex and of at most n n−1 -growth in the second variable, then suitable regularity results can be formulated. Also, SCHMIDT [65] provides an interesting alternative of a partial regularity proof for convex, fully non-autonomous integrands of (super)quadratic growth that does not utilise Gehring's lemma. The drawback here is that does not seem to generalise easily to the quasiconvex situation with (super)linear growth; even if it would, it needed to be compatible with the above proof scheme.…”

Boundary Regularity Results for Variational Integrals

“…We further deduce an immediate consequence of the previous two theorems: we first recall that in the previous two theorem we have only considered integrands f under the mixed continuity respectively Hölder continuity condition (1.3), namely on f with respect to x and on D z f with respect to (x, u). The corresponding result under the continuity respectively Hölder continuity condition (1.4) on f with respect to (x, u) now follows by an observation of Giaquinta and Giusti [32, p. 247], see also [59,Appendix A]: it was shown that if ω(·) denotes the modulus of continuity of f , then the growth condition on D zz f in (1.9) 2 allows to conclude that D z f has the corresponding (and optimal) modulus of continuity ω(·) (and no further assumption on ω is needed). Hence, we also have: In case of Theorem 1.1 it is however not clear whether the result under the assumption (1.4) follows from the one under the the assumption (1.3), since in the statement it was not required that second order derivatives of f are bounded uniformly (with a suitable growth condition in the gradient variable).…”

“…and c depends only on n, N, p, ν, L, γ and M (cf. [59,Lemma 4.3] for the convex situation in the superquadratic case). The same inequality holds true for quasi-convex functions f not depending explicitly on u, provided that the radius is sufficiently small in dependency of p, ν, L and ω 1 (·).…”

“…As explained above it is a standard technique to compare the minimizer of the original problem to a solution of an easier problem (in the sense that the solution enjoys good a priori estimates). We here compare the minimizer of the original variational functional F[ · ; B + ] locally to the solution of a linearized system following the approach of Schmidt [59] for convex functionals.…”

“…Step 1b: Approximate A-harmonicity under assumption (1.3). We only sketch briefly the modifications needed in contrast to the assumption of (1.4) in Step 1a (this is also the case discussed in detail in [59]).…”