In 1978 De Giorgi formulated the following conjecture. Let u be a solution of u = u 3 ? u in all of R n such that juj 1 and @ n u > 0 in R n. Is it true that all level sets fu = g of u are hyperplanes, at least if n 8 ? Equivalently, does u depend only on one variable? When n = 2, this conjecture was proved in 1997 by N. Ghoussoub and C. Gui. In the present paper we prove it for n = 3. The ques...

We survey existence and regularity results for semi-linear wave equations. In particular, we review the recent regularity results for the u5-Klein Gordon equation by Grillakis and this author and give a self-contained, slightly simplified proof.

We study minimizers of the energy functional

multiplicity of positive solutions for a class of semilinear problem. II. Junping Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability. multiplicity of positive solutions for a class of semilinear problems. Existence and instability of spike layer solutions to singular perturbation problems. J.

Arnold, Falk, and Winther recently showed [Bull. Amer. Math. Soc. 47 (2010), 281–354] that linear, mixed variational problems, and their numerical approximation by mixed finite element methods, can be studied using the powerful, abstract language of Hilbert complexes. In another recent article [arXiv:1005.4455], we extended the Arnold–Falk–Winther framework by analyzing variational crimes (a la...

We study the local solvability problem for a class of semilinear equations whose linear part is the Kohn Laplacian, acting on top degree forms. We also study the validity of the Poincaré lemma, in top degree, for semilinear perturbations of the tangential Cauchy-Riemann complex.

The existence, uniqueness, regularity and asymptotic behavior of global solutions of semilinear heat equations in Hilbert spaces are studied by developing new results in the theory of one-parameter strongly continuous semigroups of bounded linear operators. Applications to special semilinear heat equations in L(R) governed by pseudo-differential operators are given.