Integrability for solutions to some anisotropic obstacle problems

Abstract

This paper deals with ${{\mathcal{K}}_{\psi, \theta}^{(p_i)}}$-obstacle problems of some anisotropic elliptic equations of the type

$$\sum_{i=1}^{n} D_i (a_i(x,Du(x)))=\sum_{i=1}^{n} D_i f^i(x)$$

under some suitable coercivity and controllable growth conditions on the vector ${a(x,z)=(a_1(x,z),a_2(x,z), \ldots, a_n(x,z))}$.Assumptions on a _i (x, z) are suggested by the Euler equation of the anisotropic functional

$$\int_{\Omega} \left(\left(h+ \sum_{j=1}^n |D_ju|^{p_j}\right)^{\frac{p_1-2}{p_1}}|D_1u|^2+ \cdots + \left(h+\sum_{j=1}^n |D_ju|^{p_j}\right)^{\frac {p_n-2}{p_n}}|D_nu|^2 \right) dx.$$

We show that, higher integrability of the datum ${\theta_*=\max\{\psi, \theta\}}$ forces solutions u to have higher integrability as well.