# Remarks on a Paper by Leonetti and Siepe

## TL;DR

This paper extends previous results on higher integrability of solutions to anisotropic elliptic boundary value problems by considering obstacle problems with nonhomogeneous terms under growth and monotonicity conditions.

## Contribution

It generalizes Leonetti and Siepe's integrability results to obstacle problems with nonhomogeneous anisotropic elliptic equations.

## Key findings

- Established higher integrability of solutions under new conditions.
- Generalized previous boundary data integrability results.
- Applicable to nonhomogeneous anisotropic elliptic equations.

## Abstract

In 2012, F.Leonetti and F.Siepe [1] considered solutions to boundary value problems of some anisotropic elliptic equations of the type $$ \left\{ \begin{array}{llll} \sum\limits _{i=1}\limits^{n} D_i (a_i(x,Du(x)))=0, &x\in \Omega,\\ u(x)=\theta (x), & x\in \partial \Omega. \end{array} \right. $$ Under some suitable conditions, they obtained an integrability result, which shows that, higher integrability of the boundary datum $\theta$ forces solutions $u$ to have higher integrability as well. In the present paper, we consider ${\cal K}_{\psi,\theta}^{(p_i)}$-obstacle problems of the nonhomogeneous anisotropic elliptic equations $$ \sum_{i=1}^n D_i (a_i(x,Du(x)))=\sum_{i=1}^n D_i f^i(x). $$ Under some controllable growth and monotonicity conditions. We obtain an integrability result, which can be regarded as a generalization of the result due to Leonetti and Siepe.