On continuity of minimizers for certain quadratic growth functionals. (English) Zbl 1192.49043

Summary: In this paper we treat the regularity problem for minimizers \(u(x): \Omega\subset \mathbb R^m\to \mathbb R^n\) of quadratic growth functionals \(\int_{\Omega} A(x,u,Du)\,dx\). About the dependence on the variable \(x\) we assume only that \(A(\cdot,u,p)\) is in the class VMO as a function of \(x\). Namely, we do not assume the continuity of \(A(x,u,p)\) with respect to \(x\). We prove a partial regularity result for the case \(m\leq 4\).


49N60 Regularity of solutions in optimal control
35J50 Variational methods for elliptic systems
35B65 Smoothness and regularity of solutions to PDEs
49K40 Sensitivity, stability, well-posedness
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