Local regularity result for solutions of obstacle problems
Let \(D\) be a bounded open set, \(\psi\) a function in \(D\) and \(\theta\in W^{1,p}(D)\). For the definition of the obstacle problem associated with the second-order degenerate elliptic equation \(\nabla\cdot A(x,\nabla u(x))= 0\) with \(| A(x,\xi)|\approx|\xi|^p\), \(1< p< n\), with an obstacle \(\psi\) and boundary values \(\theta\) [see \textit{T. Kilpeläinen}, \textit{J. Heinonen} and \textit{O. Martio}, Nonlinear potential theory for degenerate elliptic equations (Clarendon Press, Oxford) (1993; Zbl 0780.31001)]. It is shown that for nonnegative obstacles of the class \(W^{1,s}_{\text{loc}}(D)\), \(p< s< n\), a solution to the obstacle problem belongs to \(L^{s^*}_{\text{loc}}(D)\) where \(s^*\) is the Sobolev conjugate of \(s\). The proof uses truncation and the DeGiorgi method. A corresponding higher integrability problem for the derivative of solutions to a similar obstacle problem has been studied in [\textit{G. Li} and \textit{O. Martio}, Ann. Acad. Sci. Fenn., Ser. A I, Math 19, No. 1, 25--34 (1994; Zbl 0803.35059)].
scientific article; zbMATH DE number 2068728