Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind

TL;DR

This paper establishes existence and uniqueness results for elliptic equations with transmission and Wentzell boundary conditions, and applies these to prove the existence of small-amplitude steady water waves with wind effects.

Contribution

It introduces new existence and uniqueness results for elliptic equations with complex boundary conditions and applies them to model steady water waves influenced by wind.

Findings

Proved Schauder estimates for elliptic equations with transmission and Wentzell boundary conditions.

Established existence of small-amplitude traveling water waves with wind and surface tension effects.

Allowed general vorticity distribution in the atmospheric region.

Abstract

In this paper, we present results about the existence and uniqueness of solutions of elliptic equations with transmission and Wentzell boundary conditions. We provide Schauder estimates and existence results in H\"older spaces. As an application, we develop an existence theory for small-amplitude two-dimensional traveling waves in an air-water system with surface tension. The water region is assumed to be irrotational and of finite depth, and we permit a general distribution of vorticity in the atmosphere.