×

Layer solutions in a half-space for boundary reactions. (English) Zbl 1102.35034

The article is concerned with the nonlinear problem \(\Delta u=0\) in \(\mathbb R^n_+\), \(\partial u/\partial \nu = f(u)\) on \(\partial \mathbb R^n_+\). The existence, uniqueness, symmetry, variational properties and asymptotic behavior of layer solutions of this problem is studied. For \(n=2\) the characterization of \(f\), for which there exists a layer solution, is given.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Alberti, Acta Appl Math 65 pp 9– (2001)
[2] Alberti, C R Acad Sci Paris Sér I Math 319 pp 333– (1994)
[3] Alberti, Arch Rational Mech Anal 144 pp 1– (1998)
[4] Ambrosio, J Amer Math Soc 13 pp 725– (2000)
[5] Amick, Acta Math 167 pp 107– (1991)
[6] ; ; Pattern formation from boundary reaction. Differential equations and dynamical systems (Lisbon, 2000), 13-18. Fields Institute Communications, 31. American Mathematical Society, Providence, 2002.
[7] Arrieta, Z Angew Math Phys 55 pp 1– (2004)
[8] Barlow, Comm Pure Appl Math 53 pp 1007– (2000)
[9] Berestycki, Ann Sc Norm Super Pisa Cl Sci (4) 25 pp 69– (1997)
[10] Berestycki, Comm Pure Appl Math 50 pp 1089– (1997)
[11] Berestycki, Duke Math J 103 pp 375– (2000)
[12] Berestycki, Bull Braz Math Soc (NS) 22 pp 1– (1991)
[13] Berestycki, Ann Sc Norm Super Pisa Cl Sci (5) 2 pp 199– (2003)
[14] ; ; Ginzburg-Landau vortices. Progress in Nonlinear Differential Equatons and Their Applications, 13. Birkhäuser Boston, Boston 1994.
[15] ; Minimizers for boundary reactions: renormalized energy, location of singularities, and applications. In preparation.
[16] Personal communication, 2001.
[17] ; Patterns in parabolic problems with nonlinear boundary conditions. Notas do ICMC/Universidade de Sao Paulo. Série Matematica, 123. Preprint, 2001.
[18] Chipot, J Math Anal Appl 223 pp 429– (1998)
[19] Cónsul, C R Acad Sci Paris Sér I Math 321 pp 299– (1995)
[20] Cónsul, J Differential Equations 157 pp 61– (1999)
[21] ; {\(\Gamma\)}-limit of a phase-field model of dislocations. Preprint, 2003.
[22] Ghoussoub, Math Ann 311 pp 481– (1998)
[23] Ghoussoub, Ann of Math (2) 157 pp 313– (2003)
[24] ; Elliptic partial differential equations of second order. Second edition. Springer, Berlin, 1983. · Zbl 0361.35003
[25] Jerison, C R Acad Sci Paris Sér I Math 333 pp 427– (2001) · Zbl 1014.35037
[26] ; Another thin-film limit of micromagnetics. Preprint. · Zbl 1074.78012
[27] A nonlocal singular perturbation problem with periodic well potential. Preprint, 2004.
[28] Boundary vortices in thin magnetic films. Preprint, 2003.
[29] Li, Duke Math J 80 pp 83– (1995)
[30] Modica, Comm Pure Appl Math 38 pp 679– (1985)
[31] Monotonicity of the energy for entire solutions of semilinear elliptic equations. Partial differential equations and the calculus of variations, vol. II, 843-850. Progress in Nonlinear Differential Equations and Their Applications, 2. Birkhäuser Boston, Boston 1989.
[32] Nonexistence theorems for semilinear elliptic problems. Doctoral dissertation. Università degli Studi di Roma ”La Sapienza,” 2002.
[33] Ni, Notices Amer Math Soc 45 pp 9– (1998)
[34] On a conjecture of De Giorgi. Preprint, 2003.
[35] Toland, J Funct Anal 145 pp 151– (1997)
[36] Toland, J Funct Anal 145 pp 136– (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.