Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 62, pp. 1-28.

Title: Higher differentiability for solutions to nonhomogeneous obstacle problems
with 1<p<2
       
Author: Zhenqiang Wang (Nankai Univ., Tianjin, China)

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<p<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 the
gradient of the obstacle ψ 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.

Submitted May 23, 2022. Published August 22, 2022.
Math Subject Classifications: 35J87, 49J40, 47J20.
Key Words: Nonhomogeneous elliptic obstacle problems; higher differentiability;
  Sobolev coefficients; Besov-Lipschitz coefficients.