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 modeling 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.
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Notes
The function \(g(x,\varvec{z})\) is called Carathéodory if it is for almost all \(x\in \Omega \) continuous with respect to \(\varvec{z}\) and also for all \(\varvec{z}\in \mathbb {R}^m\) measurable with respect to x.
We say that a couple \((\Phi ,\Psi )\) satisfies the \(\Delta _2\) condition if both functions \(\Phi \) and \(\Psi \) satisfy the \(\Delta _2\) condition.
For sake of clarity, the identity (2.12) written in terms of components of \(\varvec{w}\), \(\varvec{\varphi }\) and q has the following form
$$\begin{aligned} \sum _{i=1}^N \sum _{j=1}^d \int \limits _{\Omega }\varvec{w}_{i,j} \varvec{\varphi }_{i,j}=-\sum _{i=1}^N \int \limits _{\Omega }q_i \left( \sum _{j=1}^d \frac{\partial \varvec{\varphi }_{i,j}}{\partial x_j} \right) . \end{aligned}$$The reason for such simplification is that we do not want to employ the trace and/or the inverse trace theorem in Musielak–Orlicz spaces. But clearly, every \(\phi _D\in W^{1,\infty }(\Gamma _\mathrm{D})\) can be extended to the whole \(\Omega \) such that it satisfies the assumption (D1).
Here, in fact the function \(\psi \) depends only on the first \((d-1)\) variables, but since the set \(\Gamma _i\) is described as a graph of a Lipschitz mapping depending on \(x_1,\ldots , x_{d-1}\), we can use the standard substitution and the fundamental theorem about integrable functions.
References
Bulíček, M., Gwiazda, P., Kalousek, M., Świerczewska Gwiazda, A.: Homogenization of nonlinear elliptic systems in nonreflexive Musielak–Orlicz spaces. Nonlinearity 32(3), 1073–1110 (2019)
Bulíček, M., Gwiazda, P., Kalousek, M.: Existence and homogenization of nonlinear elliptic systems in nonreflexive spaces. Nonlinear Anal. 194, 111487 (2020). Nonlinear Potential Theory
Bulíček, M., Gwiazda, P., Málek, J., Rajagopal, K.R., Świerczewska Gwiazda, A.: On flows of fluids described by an implicit constitutive equation characterized by a maximal monotone graph, Mathematical aspects of fluid mechanics, London Mathematical Society Lecture Note Series, vol. 402. Cambridge University Press, Cambridge, pp. 23–51 (2012)
Castelli, G.F., Dörfler, W.: The numerical study of a microscale model for lithium-ion batteries. Comput. Math. Appl. 77(6), 1527–1540 (2019)
DeWitt, S., Thornton, K.: Model for anodic film growth on Aluminum with coupled bulk transport and interfacial reactions. Langmuir 30(18), 5314–5325 (2014)
Dörfler, W., Maier, M.: An elliptic problem with strongly nonlinear interface condition. Appl. Anal. 99(3), 479–495 (2020)
Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Springer Monographs in Mathematics, 2nd edn. Springer, New York (2011). Steady-state problems
Gwiazda, P., Minakowski, P., Wróblewska-Kamińska, A.: Elliptic problems in generalized Orlicz–Musielak spaces. Cent. Eur. J. Math. 10(6), 2019–2032 (2012)
Gwiazda, P., Skrzypczak, I., Zatorska-Goldstein, A.: Existence of renormalized solutions to elliptic equation in Musielak–Orlicz space. J. Differ. Equ. 264(1), 341–377 (2018)
Habera, M.: Modeling of porous metal oxide layer growth in the anodization process, Master’s thesis. Charles University, Faculty of Mathematics and Physics, Prague (2017)
Harjulehto, P., Hästö, P.: Orlicz spaces and generalized Orlicz spaces, Lecture notes in mathematics, vol. 2236, 1st edn. Springer, Cham (2019)
Houser, J.E., Hebert, K.R.: The role of viscous flow of oxide in the growth of self-ordered porous anodic alumina films. Nat. Mater. 8(5), 415–420 (2009)
Krasnosel’skiĭ, M. A., Rutickiĭ, J. B.: Convex functions and Orlicz spaces, Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen (1961)
Latz, A., Zausch, J.: Thermodynamic consistent transport theory of li-ion batteries. J. Power Sources 196(6), 3296–3302 (2011)
Less, G.B., et al.: Micro-scale modeling of Li-ion batteries: parameterization and validation. J. Electrochem. Soc. 159(6), A697–A704 (2012)
Seger, T.: Elliptic-parabolic systems with applications to Lithium-ion battery models, Ph.D. thesis, University of Konstanz, Konstanz (2013)
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This work was supported by the project No. 18-12719S financed by GAČR and by the projects No. 548217 and No. 1652119 financed by GAUK.
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Bathory, M., Bulíček, M. & Souček, O. Existence and qualitative theory for nonlinear elliptic systems with a nonlinear interface condition used in electrochemistry. Z. Angew. Math. Phys. 71, 74 (2020). https://doi.org/10.1007/s00033-020-01293-w
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DOI: https://doi.org/10.1007/s00033-020-01293-w