Convex functionals and partial regularity. (English) Zbl 0658.49005

The authors study the regularity properties of local minimizers u: \({\mathbb{R}}^ n\supset \Omega \to {\mathbb{R}}^ N\) of the functional \[ F(u,\Omega):=\int_{\Omega}f(\cdot,u,Du)dx \] with integrand f(x,y,P) convex in P and of growth order \(m\geq 1\), i.e. \[ c_ 1| P|^ m\leq f(x,y,P)\leq c_ 2\cdot (1+| P|^ m). \] For exponents \(m>1\) the minimizer is in the Sobolev space \(H^{1,m}_{loc}(\Omega,{\mathbb{R}}^ N)\), for \(m=1\) the problem is discussed in \(BV_{loc}(\Omega,{\mathbb{R}}^ N)\). In contrast to the known results no global assumptions concerning the regularity and ellipticity of the integrand are imposed, the main regularity theorem only involves a local criterion which in case of integrands \(f=f(P)\) can be summarized as follows: If u is a local F-minimizer with convex integrand f: \({\mathbb{R}}^{nN}\to {\mathbb{R}}\) and if for some points \(x_ 0\in \Omega\), \(\bar P\in {\mathbb{R}}^{nN}\) \[ \lim_{R\downarrow 0}\int_{B_ R(x_ 0)}| Du-\bar P|^ m=0 \] holds, then u is of class \(C^{1,\alpha}\) near \(x_ 0\), provided f is smooth in a neighborhood of \(\bar P\) and satisfies \(f_{p^ i_{\alpha}p^ j_{\beta}}(\bar P) Q^ i_{\alpha} Q^ j_{\beta}\leq \lambda | Q|^ 2\) with \(\lambda >0\). The proof is based on decay estimates for a quantity which measures the deviation of Du from being Hölder continuous. In order to get these inequalities one has to compare the solution u with minimizers of frozen functionals, another key ingredient entering the proof is a smoothing lemma stated in section 4 of the paper. It is worth noting that the cases \(m=1\) and \(m>1\) can be handled simultaneously with the above described unified approach.
Reviewer: M.Fuchs


49J45 Methods involving semicontinuity and convergence; relaxation
35D10 Regularity of generalized solutions of PDE (MSC2000)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
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