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

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 nanolayer 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.

A quasilinear transmission problem with application to Maxwell equations with a divergence-free $\mathcal D$-field

“…In [31] the authors analyse the case of two adjacent materials which behave according to a model with nonlinear growth of power-type in the gradient-variable ξ (which may also be different from material to material); such results have been later improved in [22,23]. More recently, quasilinear transmission problems even with a wild growth of the function a in the gradient-variable have been investigated in [7] by means of Orlicz spaces techniques. We also mention an other direction of research for transmission problems focused on the analysis of non-smooth interfaces [11].…”

Existence of a global weak solution for a reaction–diffusion problem with membrane conditions

“…. , m, where C is the same constant than in (3). Substituting in the equation for z i , we obtain that for all i = 1, .…”

Effective interface conditions for a porous medium type problem

“…1 Sorbonne Université, Inria, Université de Paris, Laboratoire Jacques-Louis Lions, UMR7598, 75005 Paris, France. Emails: ciavolella@ljll.math.upmc.fr, david@ljll.math.upmc.fr, poulain@ljll.math.upmc.fr 2 Dipartimento di Matematica, Università di Roma "Tor Vergata", Italy 3 Dipartimento di Matematica, Universitá di Bologna, Italy…”

“…When the thickness of the thin layer decreases to zero, the membrane collapses to a limiting interface, Γ1,3 , which separates two domains denoted by Ω1 and Ω3 , see Figure 1. We derive in a rigorous way the effective problem (2), and in particular, the transmission conditions on the limit density, ũ, across the effective interface. Assuming that the diffusion coefficients satisfy µ i,ε > 0 for i = 1, 3 and…”

“…The system is then closed by transmission conditions on the effective interface which generalise the classical Kedem-Katchalsky conditions. The latter were first formulated in [18] and are used to describe different diffusive phenomena, such as, for instance, the transport of molecules through the cell/nucleus membrane [9,12,34], solutes absorption processes through the arterial wall [30], the transfer of chemicals through thin biological membranes [8], or the transfer of ions through the interface between two different materials [2].…”