Pointwise regularity of solutions to nonlinear double obstacle problems
Let A and B denote respectively vector and scalar valued functions defined on \(\Omega \times R^ 1\times R^ n\) satisfying, for \(1<p<n\), \(\mu >0\) nd \(\nu\geq 0\), \[ | A(x,z,h)| \leq \mu | h|^{p- 1}+\mu | z|^{p-1}+\nu;\quad h\cdot A(x,z,h)\geq | h|^ p-\mu | z|^ p-\nu;\quad | B(x,z,h)| \leq \mu | h|^{p-1}+\mu | z|^{p-1}+\nu \] for a.e. \(x\in \Omega\) and all \(z\in R^ 1\), \(h\in R^ n\). Let \(\psi_ 1(x)\) (bounded above) and \(\psi_ 2(x)\) (bounded below) satisfy \(\psi_ 1(x)\leq \psi_ 2(x)\). A bounded function \(u\in W^{1,p}(\Omega)\) is said to be a solution of the double obstacle problem if \(\psi_ 1\leq u\leq \psi_ 2\) and \[ \int_{\Omega} A(x,u,Du)\cdot D\phi dx+\int_{\Omega} B(x,u,Du)\phi dx\geq 0 \] for all \(\phi \in W_ 0^{1,p}(\Omega)\) such that \(\psi_ 1\leq \phi \leq \psi_ 2.\) The authors show that if \(\psi_ 1\) and \(\psi_ 2\) are locally Hölder continuous in \(\Omega\) then so is u. Furthermore, they prove that if \(\psi_ 1\) and \(\psi_ 2\) satisfy a Wiener-type regularity condition then u is continuous. If A and B also satisfy A(x,u,h)\(\cdot h\geq 0\), \(A(x,u,0)=0\), \(B(x,u,0)=0\) then a modulus of continuity is obtained for u. This paper extends that of \textit{J. H. Michael} and the second author [Nonlinear Anal., Theory Methods Appl. 10, 1427-1448 (1986; Zbl 0603.49006)] in case of a single obstacle and that of \textit{G. Dal Maso}, \textit{U. Mosco} and \textit{M. A. Vivaldi} [Acta Math. 163, No.1/2, 57-107 (1989; Zbl 0696.49016)] in case of linear operators.
scientific article; zbMATH DE number 4212574
scientific article