Existence and blow up of solutions to certain classes of two-dimensional nonlinear Neumann problems

Medville, Kai; Vogelius, Michael S.

doi:10.1016/j.anihpc.2005.02.008

Medville, Kai ; Vogelius, Michael S.

Annales de l'I.H.P. Analyse non linéaire, Tome 23 (2006) no. 4, pp. 499-538

MR Zbl

DOI : 10.1016/j.anihpc.2005.02.008

@article{AIHPC_2006__23_4_499_0,
     author = {Medville, Kai and Vogelius, Michael S.},
     title = {Existence and blow up of solutions to certain classes of two-dimensional nonlinear {Neumann} problems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {499--538},
     year = {2006},
     publisher = {Elsevier},
     volume = {23},
     number = {4},
     doi = {10.1016/j.anihpc.2005.02.008},
     mrnumber = {2245754},
     zbl = {05060815},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2005.02.008/}
}

TY  - JOUR
AU  - Medville, Kai
AU  - Vogelius, Michael S.
TI  - Existence and blow up of solutions to certain classes of two-dimensional nonlinear Neumann problems
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2006
SP  - 499
EP  - 538
VL  - 23
IS  - 4
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.anihpc.2005.02.008/
DO  - 10.1016/j.anihpc.2005.02.008
LA  - en
ID  - AIHPC_2006__23_4_499_0
ER  -

%0 Journal Article
%A Medville, Kai
%A Vogelius, Michael S.
%T Existence and blow up of solutions to certain classes of two-dimensional nonlinear Neumann problems
%J Annales de l'I.H.P. Analyse non linéaire
%D 2006
%P 499-538
%V 23
%N 4
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.anihpc.2005.02.008/
%R 10.1016/j.anihpc.2005.02.008
%G en
%F AIHPC_2006__23_4_499_0

Medville, Kai; Vogelius, Michael S. Existence and blow up of solutions to certain classes of two-dimensional nonlinear Neumann problems. Annales de l'I.H.P. Analyse non linéaire, Tome 23 (2006) no. 4, pp. 499-538. doi: 10.1016/j.anihpc.2005.02.008

Bibliographie
Cité par

[1] Ando T., Fowler A.B., Stern F., Electronic properties of two-dimensional system, Rev. Modern Phys. 54 (1982) 437-621.

[2] Bebernes J., Eberly D., Mathematical Problems from Combustion Theory, Appl. Math. Sci., vol. 83, Springer-Verlag, Berlin, 1989. | Zbl | MR

[3] Bethuel F., Brezis H., Helein F., Ginzburg-Landau Vortices, Birkhäuser, Boston, 1994. | Zbl | MR

[4] Brezis H., Merle F., Uniform estimates and blow-up behavior for solutions of $- Δ u = V (x) e^{u}$ in two dimensions, Comm. Partial Differential Equations 16 (1991) 1223-1253. | Zbl | MR

[5] Bryan K., Vogelius M., Singular solutions to a nonlinear elliptic boundary value problem originating from corrosion modeling, Quart. Appl. Math. 60 (2002) 675-694. | Zbl | MR

[6] J. Davila, A linear elliptic equation with a nonlinear boundary condition, Preprint, Rutgers University, 2001.

[7] Deconinck J., Current Distributions and Electrode Shape Changes in Electrochemical Systems, Lecture Notes in Engrg., vol. 75, Springer-Verlag, Berlin, 1992.

[8] Jacobsen J., A globalization of the Implicit Function Theorem with applications to nonlinear elliptic equations, Contemp. Math. 289 (2001) 249-272. | MR | Zbl

[9] Kavian O., Introduction à la théorie des points critiques et applications aux problèmes elliptiques, Math. Appl., vol. 13, Springer-Verlag, Berlin, 1993. | Zbl | MR

[10] Kavian O., Vogelius M., On the existence and “blow up” of solutions to a two-dimensional nonlinear boundary-value problem arising in corrosion modelling, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 119-149, Corrigendum to same, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 729-730. | Zbl

[11] K. Medville, Ph.D. Thesis, Rutgers University, 2004.

[12] Medville K., Vogelius M.S., Blow-up behavior of planar harmonic functions satisfying a certain exponential Neumann boundary condition, SIAM J. Math. Anal. 36 (2005) 1772-1806. | Zbl | MR

[13] Nagasaki K., Suzuki T., Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities, Asymptotic Anal. 3 (1990) 173-188. | Zbl | MR

[14] Rabinowitz P., Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. in Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986. | Zbl | MR

[15] Shamma S.E., Asymptotic behavior of Stekloff eigenvalues and eigenfunctions, SIAM J. Appl. Math. 20 (1971) 482-490. | Zbl | MR

[16] Struwe M., Variational Methods, Ergeb. Math. Grenzgeb., vol. 34, Springer-Verlag, Berlin, 1996. | Zbl | MR

[17] Vogelius M., Xu J.-M., A nonlinear elliptic boundary value problem related to corrosion modeling, Quart. Appl. Math. 56 (1998) 479-505. | Zbl | MR

Cité par Sources :