Partial regularity for variational integrals with \((s,\mu,q)\)-growth. (English) Zbl 1018.49026

Bildhauer, Michael; Fuchs, Martin

Calc. Var. Partial Differ. Equ. 13, No. 4, 537-560 (2001).

The authors consider the vector integral functional \[ J(u) = \int_\Omega f(Du) dx \] when \(f\in C^2(\mathbb{R}^{nN})\) (where \(\mathbb{R}^{nN}\) denotes the set of all \(n\times N\) matrices) satisfies suitable growth conditions depending on three real parameters \(s \geq 1\), \(q>1\) and \(\mu\) such that \(q<2-\mu+2s/n\). Specifically, they assume first that there is a function \(F\colon [0,\infty)\to [0,\infty)\) such that \[ \lim_{t\to\infty} \frac {F(t)}t =\infty, \quad \liminf_{t\to\infty} \frac {F(t)}{t^s} >0 \] and \[ \inf_{Z\in \mathbb{R}^{nN}} \frac {f(Z)}{F(|Z|)} >0. \] They also assume that there are positive constants such that \[ \lambda(1+|Z|^2)^{-\mu/2} |Y|^2 \leq D^2(Z)(Y,Y) \leq \Lambda \lambda(1+|Z|^2)^{(q-2/2} |Y|^2 \] for all \(Z\) and \(Y\) in \(\mathbb{R}^{nN}\). Since \(u\) is a vector-valued function, one cannot expect it to be globally smooth. Instead, the authors prove a partial regularity result, namely, that, for any local minimizer, \(u\), of \(J\) there is a closed subset \(K\) of \(\Omega\) with zero measure such that \(u\) has Hölder continuous derivatives on \(\Omega\setminus K\).
Examples are given to illustrate \((s,\mu,q)\) growth, with the model case being \(f(Z)=|Z|\ln(1+ |Z|)\), with \(s=\mu=1\) and \(q=1+\varepsilon\) for some \(\varepsilon \in (0,n/2)\) and \(F(t) = t\ln(1+t)\). Other examples demonstrate the relation between this result and other similar ones. The proof is via a blow-up argument.

Reviewer: Gary M.Lieberman (Ames)

MSC:

49N60	Regularity of solutions in optimal control
35J45	Systems of elliptic equations, general (MSC2000)

Keywords:

partial regularity; anisotropic growth conditions; vector integral functional; vector-valued function