The space charge problem and the nonlinear resistor
Author:
Giovanni Cimatti
Journal:
Quart. Appl. Math. 74 (2016), 49-59
MSC (2010):
Primary 35J66, 35J75, 78M35
DOI:
https://doi.org/10.1090/qam/1422
Published electronically:
December 3, 2015
MathSciNet review:
3472519
Full-text PDF Free Access
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Abstract: We propose the constitutive equation $\mathbf {J}=f(q)\mathbf {E}$, where $\mathbf {J}$ is the current density, $q$ the charge density, $\mathbf {E}$ the electric field and $f(q)$ a given function to describe the current-voltage laws appearing in certain special materials. A theorem of existence and uniqueness of solution is also given for the related nonlinear boundary value problem.
References
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- T. Bhattacharya, A note on the space charge problem, IMA J. Appl. Math. 48 (1992), no. 2, 117–124. MR 1159834, DOI 10.1093/imamat/48.2.117
- Chris J. Budd, Avner Friedman, Bryce McLeod, and Adam A. Wheeler, The space charge problem, SIAM J. Appl. Math. 50 (1990), no. 1, 181–198. MR 1036238, DOI 10.1137/0150012
- C. J. Budd and A. A. Wheeler, Exact solutions of the space charge equation using the hodograph method, IMA J. Appl. Math. 40 (1988), no. 1, 1–14. MR 983746, DOI 10.1093/imamat/40.1.1
- Giovanni Cimatti, Existence of weak solutions for the space-charge problem, IMA J. Appl. Math. 44 (1990), no. 2, 185–195. MR 1052510, DOI 10.1093/imamat/44.2.185
- D. Darnall, A. M. Stoneham, and D. V. Morgan, Electrical phenomena in amorphous films, Rep. Prig. Phys. 33 (1970), 1129-1191.
- W. Detach, Ümber die Dichtverteilung unipolarer Ionenströme, Ann. Phys. 16 (1933).
- C. B. Duke, Tunneling in solids, Academic Press, New York and London, 1969.
- N. Felici, Recent advances in analysis of DC ionized electric fields, I, Direct Currents, September 1963, 252-260.
- N. Felici, Recent advances in analysis of DC ionized electric fields, II, Direct Currents, October 1963, 278-287.
- H. Fritzsche and Stanford R. Ovshinsky, A sketch, Phys. Status Solidi B 249 (2012), 1827-1830.
- T. W. Hickmott, Low-frequency negative resistance in thin anodic oxide films, J. Appl. Phys. 33 (1962), 2669-2682.
- A. A. Milgram and C. Lu, Field effect and electrical conduction mechanism in discontinuous thin metal films, J. Appl. Phys. 37 (1966), 4773-4779.
- D. B. Strukov, G. S. Sneider, D. R. Stewart, and R. S. Williams, The missing memristor found, Nature 453 (2008), 80-83.
- Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
References
- Pierre Atten, Existence et unicité de la solution de l’équation des champs électriques affectés par un flux permanent d’ions unipolaires, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), B1188–B1190 (French). MR 0233554 (38 \#1875)
- T. Bhattacharya, A note on the space charge problem, IMA J. Appl. Math. 48 (1992), no. 2, 117–124. MR 1159834 (93a:78004), DOI 10.1093/imamat/48.2.117
- Chris J. Budd, Avner Friedman, Bryce McLeod, and Adam A. Wheeler, The space charge problem, SIAM J. Appl. Math. 50 (1990), no. 1, 181–198. MR 1036238 (91b:78004), DOI 10.1137/0150012
- C. J. Budd and A. A. Wheeler, Exact solutions of the space charge equation using the hodograph method, IMA J. Appl. Math. 40 (1988), no. 1, 1–14. MR 983746 (90c:78010), DOI 10.1093/imamat/40.1.1
- Giovanni Cimatti, Existence of weak solutions for the space-charge problem, IMA J. Appl. Math. 44 (1990), no. 2, 185–195. MR 1052510 (91f:35074), DOI 10.1093/imamat/44.2.185
- D. Darnall, A. M. Stoneham, and D. V. Morgan, Electrical phenomena in amorphous films, Rep. Prig. Phys. 33 (1970), 1129-1191.
- W. Detach, Ümber die Dichtverteilung unipolarer Ionenströme, Ann. Phys. 16 (1933).
- C. B. Duke, Tunneling in solids, Academic Press, New York and London, 1969.
- N. Felici, Recent advances in analysis of DC ionized electric fields, I, Direct Currents, September 1963, 252-260.
- N. Felici, Recent advances in analysis of DC ionized electric fields, II, Direct Currents, October 1963, 278-287.
- H. Fritzsche and Stanford R. Ovshinsky, A sketch, Phys. Status Solidi B 249 (2012), 1827-1830.
- T. W. Hickmott, Low-frequency negative resistance in thin anodic oxide films, J. Appl. Phys. 33 (1962), 2669-2682.
- A. A. Milgram and C. Lu, Field effect and electrical conduction mechanism in discontinuous thin metal films, J. Appl. Phys. 37 (1966), 4773-4779.
- D. B. Strukov, G. S. Sneider, D. R. Stewart, and R. S. Williams, The missing memristor found, Nature 453 (2008), 80-83.
- Olga A. Ladyzhenskaya and Nina N. Ural′tseva, Linear and quasilinear elliptic equations, translated from the Russian by Scripta Technica, Inc., translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. MR 0244627 (39 \#5941)
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Additional Information
Giovanni Cimatti
Affiliation:
Department of Mathematics, Largo Bruno Pontecorvo, 5, 56100 Pisa, Italy
Email:
cimatti@dm.unipi.it
Received by editor(s):
April 18, 2014
Received by editor(s) in revised form:
October 22, 2014
Published electronically:
December 3, 2015
Article copyright:
© Copyright 2015
Brown University
