# Existence and Regularity of Weak Solutions for a Thermoelectric Model

## TL;DR

This paper proves the existence and regularity of weak solutions for a nonlinear, coupled thermoelectric model involving Maxwell and elliptic equations, under general boundary conditions on Lipschitz domains.

## Contribution

It introduces a novel analysis combining De Giorgi-Nash and Campanato methods to establish existence and regularity results for a complex thermoelectric system.

## Key findings

- Existence of weak solutions on Lipschitz domains.
- Regularity results for the weak solutions.
- Applicability to general boundary conditions.

## Abstract

This paper concerns a time-independent thermoelectric model with two different boundary conditions. The model is a nonlinear coupled system of the Maxwell equations and an elliptic equation. By analyzing carefully the nonlinear structure of the equations, and with the help of the De Giorgi-Nash estimate for elliptic equations, we obtain existence of weak solutions on Lipschitz domains for general boundary data. Using Campanato's method, we establish regularity results of the weak solutions.