Assume that f ( s ) = F ′ ( s ) where F is a double-well potential. Under certain conditions on the Lipschitz constant of f on [ − 1 , 1 ] , we prove that arbitrary bounded global solutions of the semilinear equation Δ u = f ( u ) on hyperbolic space H n must reduce to functions of one variable provided they admit asymptotic boundary values on S n − 1 = ∂ ∞ H n which are invariant under a cohomogeneity one subgroup of the group of isometries of H n . We also prove existence of these one-dimensional solutions.
Symmetry for Solutions of Two-phase Semilinear Elliptic Equations on Hyperbolic Space / Birindelli, Isabella; Rafe, Mazzeo. - In: INDIANA UNIVERSITY MATHEMATICS JOURNAL. - ISSN 0022-2518. - 58:5(2009), pp. 2347-2368. [10.1512/iumj.2009.58.3714]
Symmetry for Solutions of Two-phase Semilinear Elliptic Equations on Hyperbolic Space
BIRINDELLI, Isabella;
2009
Abstract
Assume that f ( s ) = F ′ ( s ) where F is a double-well potential. Under certain conditions on the Lipschitz constant of f on [ − 1 , 1 ] , we prove that arbitrary bounded global solutions of the semilinear equation Δ u = f ( u ) on hyperbolic space H n must reduce to functions of one variable provided they admit asymptotic boundary values on S n − 1 = ∂ ∞ H n which are invariant under a cohomogeneity one subgroup of the group of isometries of H n . We also prove existence of these one-dimensional solutions.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.