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

Ragusa, Maria Alessandra; Tachikawa, Atsushi

J. Math. Soc. Japan 57, No. 3, 691-700 (2005).

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 \(m\leq 4\).

MSC:

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

Keywords:

variational problem; minimizer; partial regularity