Existence and qualitative theory for nonlinear elliptic systems with a nonlinear interface condition used in electrochemistry

TL;DR

This paper establishes the existence of solutions for nonlinear elliptic systems with interface conditions relevant in electrochemistry, accommodating both rapid and slow growth nonlinearities within the model.

Contribution

It provides a new existence theory for nonlinear elliptic systems with discontinuous constitutive laws, under delta-two or nabla-two growth conditions.

Findings

Existence of solutions under delta-two or nabla-two conditions.

Applicable to models with exponential and logarithmic growth.

Addresses regularity issues in nonlinear interface problems.

Abstract

We study a nonlinear elliptic system with prescribed inner interface conditions. These models are frequently used in physical system where the ion transfer plays the important role for example in modelling of nano-layer growth or Li-on batteries. The key difficulty of the model consists of the rapid or very slow growth of nonlinearity in the constitutive equation inside the domain or on the interface. While on the interface, one can avoid the difficulty by proving a kind of maximum principle of a solution, inside the domain such regularity for the flux is not available in principle since the constitutive law is discontinuous with respect to the spatial variable. The key result of the paper is the existence theory for these problems, where we require that the leading functional satisfies either the delta-two or the nabla-two condition. This assumption is applicable in case of fast (exponential) growth as well as in the case of very slow (logarithmically superlinear) growth.