Heteroclinic solutions of Allen-Cahn type equations with a general elliptic operator
Keywords
Heteroclinic solutions, Allen-Cahn equationAbstract
We consider a generalization of the Allen-Cahn type equation in divergence form $-\rom{div}(\nabla G(\nabla u(x,y)))+F_u(x,y,u(x,y))=0$. This is more general than the usual Laplace operator. We prove the existence and regularity of heteroclinic solutions under standard ellipticity and $m$-growth conditions.References
F. Alessio, L. Jeanjean and P. Montecchiari, Stationary layered solutions in R2 for a class of non autonomous Allen–Cahn equations, Calc. Var. Partial Differential Equations 11 (2000), 177–202.
F. Alessio, L. Jeanjean and P. Montecchiari, Existence of infinitely many stationary layered solutions in R2 for a class of periodic Allen–Cahn equations, Com. Partial Differential Equations 27 (2002), 1537–1574.
V. Bangert, On minimal laminations of the torus, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), 95–138.
U. Bessi, Slope-changing solutions of elliptic problems on Rn , Nonlinear Anal. 68 (2008), 3923–3947.
B. Dacorogna, Direct Methods in the Calculus of Variations, Springer, New York, 2007.
R. de la Llave and E. Valdinoci, Multiplicity results for interfaces of Ginzburg–Landau–Allen–Cahn equations in periodic media, Adv. Math. 215 (2007), 379–426.
M. Giaquinta and E. Giusti, On the regularity of the minima of variatonal integrals, Acta Math. 148 (1982), 31–46.
O.A. Ladyzhenskaya and N.N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), 229–272.
P. Pucci and J.B. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007.
P.H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen–Cahn type equation, Comm. Pure Appl. Math. 56 (2003), 1078–1134.
E. Valdinoci, Plane-like minimizers in periodic media: jet flows and Ginzburg–Landautype functionals, J. Reine Angew. Math. 574 (2001), 147–185.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
