Partial regularity of minimizers of quasiconvex integrals with subquadratic growth
Given a domain \(\Omega\) in \(\mathbb{R}^n\) and an exponent \(p> 1\), the authors investigate the regularity properties of local minimizers \(u\in W^1_p(\Omega, \mathbb{R}^n)\) of the variational integral \[ I(w)= \int_\Omega f(Dw) dx \] with integrand \(f\in C^2(\mathbb{R}^{nN})\) being of \(p\)-growth and satisfying in addition the condition of uniform strict quasiconvexity. Starting with the work of \textit{L. C. Evans} [Arch. Ration. Mech. Anal. 95, 227-252 (1986; Zbl 0627.49006)] and with further refinements of \textit{E. Acerbi} and \textit{N. Fusco} [Arch. Ration. Mech. Anal. 99, 261-281 (1987; Zbl 0627.49007)] it is known that in the case of exponents \(p\geq 2\) local minimizers are of class \(C^{1,\alpha}\) on an open subset of \(\Omega\) with full Lebesgue measure. In [Proc. R. Soc. Lond., Ser. A 433, No. 1889, 723-725 (1991; Zbl 0741.49016)] \textit{V. Šverak} constructed a quasiconvex (and not convex neither polyconvex) integrand \(f\) of growth order \(1< p< 2\) so that it makes strong sense to discuss regularity also in the subquadratic case. Using blow-up arguments the authors succeed in establishing partial regularity of minimizers in the above described sense. Although the general line of argumentation is quite familiar to experts, it should be noted that the case \(p< 2\) requires various new techniques, for example, the authors use a variant of the Sobolev-Poincaré inequality on balls with increasing radius and they prove a regularity result for elliptic systems with constant coefficients saying that any weak solution of class \(W^1_1\) in fact is smooth.
scientific article; zbMATH DE number 1471598