# A comparison theorem for nonsmooth nonlinear operators

## TL;DR

This paper establishes a comparison theorem for nonsmooth nonlinear PDE operators with applications to estimating periodic water wave profiles, based on advanced maximum principles and elliptic equation theory.

## Contribution

It introduces a novel comparison theorem for nonsmooth nonlinear operators with non-vanishing gradients, extending PDE analysis techniques.

## Key findings

- Proves a comparison theorem for super- and sub-solutions with non-vanishing gradients.
- Develops a maximum principle for divergence type elliptic equations with VMO coefficients.
- Applies results to estimate periodic water wave profiles.

## Abstract

We prove a comparison theorem for super- and sub-solutions with non-vanishing gradients to semilinear PDEs provided a nonlinearity $f$ is $L^p$ function with $p > 1$. The proof is based on a strong maximum principle for solutions of divergence type elliptic equations with VMO leading coefficients and with lower order coefficients from a Kato class. An application to estimation of periodic water waves profiles is given.