A nonlinear elliptic boundary value problem related to corrosion modeling
Authors:
Michael Vogelius and Jian-Ming Xu
Journal:
Quart. Appl. Math. 56 (1998), 479-505
MSC:
Primary 35J65; Secondary 35Q72, 73B99, 73C99
DOI:
https://doi.org/10.1090/qam/1637048
MathSciNet review:
MR1637048
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Abstract: We study a nonlinear boundary value problem arising from electrochemistry. The essential difficulties are due to the strong nonlinear nature of part of the boundary condition. This part of the boundary condition is of an exponential type and is normally in the corrosion literature associated with the names of Butler and Volmer. We examine the questions of existence and uniqueness of solutions to this boundary value problem. In a numerical example we compare the behaviour of the solutions to the nonlinear problem with the behaviour of the solutions to a corresponding linearized problem. In contrast to earlier studies we put a major emphasis on studying parameter values that may be relevant for the case in which part of the boundary is in a transition to passivity—in practice most likely because it is nearly covered by an oxide layer.
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R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975
R. Baboian, Corrosion—A Problem of International Importance, Corrosion Testing and Evaluation: Silver Anniversary Volume, ASTM STP 1000, R. Baboian and S. S. Dean, eds., American Society for Testing and Materials, Philadelphia, 1990, pp. 7–13.
J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, Berlin, 1976.
H. Brezis and L. Nirenberg, book in preparation
J. Deconinck, Current Distributions and Electrode Shape Changes in Electrochemical Systems, Lecture Notes in Engineering, Vol. 75, Springer-Verlag, Berlin, 1992
Z.-C. Han, Prescribing Gaussian curvature on $S^{2}$, Duke Mathematical Journal 61, 679–703 (1990)
L. S. Hou and J. C. Turner, Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls, Numer. Math. 71, 289–315 (1995)
M. A. Krasnosel’skii and Ya. B. Rutickii, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961
D. Landolt, Electrochemical and materials science aspects of alloy deposition, Electrochimica Acta 39, 1075–1090 (1994)
M. Matlosz, C. Creton, C. Clerc, and D. Landolt, Secondary current distribution in a Hull cell, J. Electrochem. Soc. 134, 3015–3021 (1987)
J. Moser, On a nonlinear problem in differential geometry, In Dynamical Systems, M. Peixoto, Editor, Academic Press, New York, 1973
J. S. Newman, Electrochemical Systems, Prentice-Hall, Englewood Cliffs, NJ, 1973
J. Peetre, Espaces d’Interpolation et Théorème de Soboleff, Ann. Inst. Fourier (Grenoble) 16, 279–317 (1966)
P. Ponthiaux, F. Wenger, and J. Galland, Study of the anodic current-voltage curve of an iron-nickel alloy in normal sulfuric acid, J. Electrochem. Soc. 142, 2204–2210 (1966)
N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17, 473–483 (1967)
H. H. Uhlig and R. W. Revie, Corrosion and Corrosion Control, John Wiley and Sons, New York, 1985
N. G. Zamani, J. F. Porter, and A. A. Mufti, A survey of computational efforts in the field of corrosion engineering, International Journal for Numerical Methods in Engineering 23, 1295–1311 (1986)
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Article copyright:
© Copyright 1998
American Mathematical Society