Convex functionals and partial regularity
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.
scientific article; zbMATH DE number 4075921
scientific article