Journal of the
Korean Mathematical Society JKMS

We consider regularity questions arising in the degenerate elliptic vector valued variational inequalities \begin{equation*} -\text{div} (\vert\nabla u\vert^{p-2} \nabla u) \geq b(x,u,\nabla u) \end{equation*} with $p\in (1,\infty)$. It is a generalization of the scalar valued inequalities, i.e., the obstacle problem. We obtain the $C_{\text{loc}}^{1,\alpha}$ regularity for the solution $u$ under a controllable growth condition of $b(x, u, \nabla u)$.

Keywords: vector valued variational inequalities, regularity, maximum principle

MSC numbers: 35J85, 35B99, 58E35