Boundedness of weak solutions of degenerate quasilinear equations with rough coefficients

TL;DR

This paper establishes local boundedness estimates for weak solutions of a broad class of degenerate quasilinear equations with rough coefficients, extending classical results to more general and less smooth cases.

Contribution

It introduces new local boundedness estimates for weak solutions of degenerate quasilinear equations with minimal structural assumptions and rough coefficients, generalizing previous classical results.

Findings

Derived local boundedness estimates for weak solutions

Extended classical results to equations with rough coefficients

Included degenerate and subelliptic equations in the analysis

Abstract

We derive local boundedness estimates for weak solutions of a large class of second order quasilinear equations. The structural assumptions imposed on an equation in the class allow vanishing of the quadratic form associated with its principal part and require no smoothness of its coefficients. The class includes second order linear elliptic equations as studied by D. Gilbarg and N. S. Trudinger [1998] and second order subelliptic linear equations as studied by E. Sawyer and R. L. Wheeden [2006 and 2010]. Our results also extend ones obtained by J. Serrin [1964] concerning local boundedness of weak solutions of quasilinear elliptic equations.