Higher differentiability for solutions to nonhomogeneous obstacle problems with 1<p<2

Abstract

In this article, we establish integer and fractional higher-order differentiability of weak solutions to non-homogeneous obstacle problems that satisfy the variational inequality $$ \int_{\Omega} \langle A(x,Du),D(\varphi-u)\rangle\,dx \ge \int_{\Omega} \langle |F|^{p-2}F,D(\varphi-u)\rangle\,dx, $$where \(1&lt; p&lt;2\), \(\varphi \in \mathcal{K}_{\psi } (\Omega ) =\{ v\in u_0+W_0^{1,p}(\Omega ,\mathbb{R} ):v\ge \psi \text{ a.e.\ in } \Omega\} \), \)(u_0\in W^{1,p}(\Omega)\) is a fixed boundary datum.We show that the higher differentiability of integer or fractional order of thegradient of the obstacle \(\psi\) and the nonhomogeneous term F can transfer to the gradient of the weak solution, provided the partial map \(x\mapsto A(x,\xi)\)belongs to a suitable Sobolev or Besov-Lipschitz space.