entire solutions of semilinear elliptic equations in r 3 and a conjecture of de giorgi this paper is concerned with the study of bounded solutions of semilinear elliptic equations u f u in the whole space r under the assumption that u is monotone in one direction say nu in r n the goal is to establish the one dimensional character or symmetry of u namely that u only depends on one variable or equivalently that the level sets of u are hyperplanes this type of symmetry question was raised by de giorgi in who made the following conjecture we quote literally page of dg

solutions of nonlinear dirac equations abstract we study the dirac equation i t i c k 1 3 k k m c 2 g x and obtain existence and multiplicity results of stationary solutions for several classes of nonlinearities g r 3 c 4 r modeling various types of interaction a typical result states that if g x u depends periodically on x and is even in u the problem has infinitely many geometrically different localized solutions the arguments are variational the associated lagrangian functional is strongly indefinite and the palais smale condition does not hold we apply some recently developed critical point theorems

ground state solutions for nonperiodic dirac equation with superquadratic nonlinearity this paper is concerned with the following nonperiodic dirac equation i k 13 k ku a u m x u fu x u in r3 where m x is a vector potential under weak superquadratic condition on the nonlinearity we establish the existence of ground state solutions of nehari type by using the generalized nehari manifold method developed recently by szulkin and weth for above problem

equivariant minimax and minimal surfaces in geometric three manifolds minimal surfaces in riemannian three dimensional manifolds have played a major role in recent studies of the geometry and topology of 3 manifolds in particular stable and least area minimal surfaces have been used extensively sy msy hs on the other hand explicit examples of unstable minimal surfaces have rarely been given for the 3 sphere s with the standard metric of constant sectional curvature there is the classical paper of lawson lh showing that closed orientable surfaces of every genus occur as unstable minimal surfaces embedded in 5 3 more recently karcher pinkall and sterling kps have constructed several new examples in s we have discovered new infinite families of minimal surfaces in s more generally we announce a number of new finite and infinite families of embedded minimal surfaces in geometric 3 manifolds using the minimax procedure described below geometric structures on 3 manifolds were introduced by thurston tw see also the excellent survey by scott sp1 there are eight geometries r s s x r nil h 2 x r s l r solv and h a geometric structure on a 3 manifold s is a representation of e as a quotient of one of the above eight spaces divided out by a covering transformation group acting isometrically equivalently e is locally isometric to one of these spaces with its natural homogeneous space structure the first six of these geometries give seifert fiber spaces and we are mainly interested in such examples the so 2 isometry actions associated with most seifert fiber spaces yield infinite classes of embedded minimal surfaces note that interesting examples can also be obtained in the other two geometries e g in hyperbolic geometry h 3 pr3 2 the minimax procedure has proved to be a versatile and powerful means of constructing unstable minimal surfaces in 3 manifolds for basic details of this technique see pj ss p r l and pr2 suppose that g is a finite group of isometries acting on a closed oriented riemannian 3 manifold e assume that a is a heegaard surface in e i e the closures of the components of e a are handlebodies k and k assume furthermore that a is g equivariant i e ga a for all g e g we consider one parameter smooth families at t 0 1 sweeping out e and having the following properties ao and ai are graphs at is isotopic to a for all 0 t 1 at is g equivariant for all t the handlebody kt is chosen so that the orientation on at is induced from that on

existence of localized solutions for a classical nonlinear dirac field we prove the existence of stationary states for nonlinear dirac equations of the form i gamma mu partial mu psi m psi f bar psi psi psi 0 we seek solutions which are separable in spherical coordinates and we use a shooting method for solving the associated problem of ordinary differential equations

the liouville property and a conjecture of de giorgi we consider bounded entire solutions of the nonlinear pde u u u 3 0i n r d and prove that under certain monotonicity conditions these solutions must be constant on hyperplanes the proof uses a liouville theorem for harmonic functions associated with a nonuniformly elliptic divergence form operator c 2000 john wiley sons inc

stable solutions of the allen cahn equation in dimension 8 and minimal cones abstract for all n 1 we are interested in bounded solutions of the allen cahn equation u u u 3 0 which are defined in all r n 1 and whose zero set is asymptotic to a given minimal cone in particular in dimension n 1 8 we prove the existence of stable solutions of the allen cahn equation whose zero sets are not hyperplanes

infinitely many solutions for a class of nonlinear dirac equations without symmetry abstract this paper is concerned with the multiplicity of solutions for the nonlinear dirac equations i k 1 3 k k u v x a u u f u x u without symmetry assumption on f we establish the existence of infinitely many geometrically distinct solutions for superquadratic as well as asymptotically quadratic nonlinearities via variational approach

minimal surfaces in the three sphere by doubling the clifford torus we construct embedded closed minimal surfaces in the round three sphere bbb s 3 1 resembling two parallel copies of the clifford torus joined by m 2 small catenoidal bridges symmetrically arranged along a square lattice of points on the torus

from constant mean curvature hypersurfaces to the gradient theory of phase transitions given a nondegenerate minimal hypersurface in a riemannian manifold we prove that for all e small enough there exists ue a critical point of the allen cahn energy ee u e2 u 2 1 u2 2 whose nodal set converges to as e tends to 0 moreover if is a volume nondegenerate constant mean curvature hypersurface then the same conclusion holds with the function ue being a critical point of ee under some volume constraint

ground state solutions for some indefinite variational problems abstract we consider the nonlinear stationary schrodinger equation u v x u f x u in r n here f is a superlinear subcritical nonlinearity and we mainly study the case where both v and f are periodic in x and 0 belongs to a spectral gap of v inspired by previous work of li et al 2006 11 and pankov 2005 13 we develop an approach to find ground state solutions i e nontrivial solutions with least possible energy the approach is based on a direct and simple reduction of the indefinite variational problem to a definite one and gives rise to a new minimax characterization of the corresponding critical value our method works for merely continuous nonlinearities f which are allowed to have weaker asymptotic growth than usually assumed for odd f we obtain infinitely many geometrically distinct solutions the approach also yields new existence and multiplicity results for the dirichlet problem for the same type of equations in a bounded domain

regularity of flat level sets in phase transitions we consider local minimizers of the ginzburg landau energy functional 1 2 u 2 1 4 1 u 2 2 dx and prove that if the 0 level set is included in a flat cylinder then in the interior it is included in a flatter cylinder as a consequence we prove a conjecture of de giorgi which states that level sets of global solutions of u u 3 u such that u 1 n u 0 lim u x x n 1 x n are hyperplanes in dimension n 8

on de giorgi s conjecture in dimensions 4 and 5 in this paper we develop an approach for establishing in some important cases a conjecture made by de giorgi more than 20 years ago the problem originates in the theory of phase transition and is so closely connected to the theory of minimal hypersurfaces that it is sometimes referred to as the eversion of bernstein s problem for minimal graphs the conjecture has been completely settled in dimension 2 by the authors 15 and in dimension 3 in 2 yet the approach in this paper seems to be the first to use in an essential way the solution of the bernstein problem stating that minimal graphs in euclidean space are necessarily hyperplanes provided the dimension of the ambient space is not greater than 8 we note that the solution of bernstein s problem was also used in 18 to simplify an argument in 9 here is the conjecture as stated by de giorgi 12 conjecture 1 1 suppose that u is an entire solution of the equation

localization of andre quillen goodwillie towers and the periodic homology of infinite loopspaces abstract let k n be the n th morava k theory at a prime p and let t n be the telescope of a v n self map of a finite complex of type n in this paper we study the k n homology of x the 0th space of a spectrum x and many related matters we give a sampling of our results let p x be the free commutative s algebra generated by x it is weakly equivalent to the wedge of all the extended powers of x we construct a natural map s n x l t n p x l t n x of commutative algebras over the localized sphere spectrum l t n s the induced map of commutative cocommutative k n hopf algebras s n x k n p x k n x satisfies the following properties it is always monic it is an isomorphism if x is n connected n 1 x is torsion and t i x 0 for 1 i n 1 it is an isomorphism only if k i x 0 for 1 i n 1 it is universal the domain of s n x preserves k n isomorphisms and if f is any functor preserving k n isomorphisms then any natural transformation f x k n x factors uniquely through s n x the construction of our natural transformation uses the telescopic functors constructed and studied previously by bousfield and the author and thus depends heavily on the nilpotence theorem of devanitz hopkins and smith our proof that s n x is always monic uses topological andre quillen homology and goodwillie calculus in nonconnective settings

on entire solutions of an elliptic system modeling phase separations we study the qualitative properties of a limiting elliptic system arising in phase separation for bose einstein condensates with multiple states

one dimensional symmetry of bounded entire solutions of some elliptic equations this paper is about one dimensional symmetry properties for some entire and bounded solutions of u f u 0 in ir we consider solutions u such that 1 u 1 and u x1 xn 1 as xn uniformly with respect to x1 xn 1 under some conditions on f we prove that the solutions only depend on the variable xn we also discuss more general elliptic operators the qualitative properties then strongly depend on the coefficients of the operator these results extend to higher dimensions and to more general operators a result of ghoussoub and gui 21 proved for n 3 ams classification 35b05 35b40 35b50 35j60

two end solutions to the allen cahn equation in r3 abstract in this paper we study axially symmetric solutions of allen cahn equation in the three dimensional euclidean space using a sophisticated continuation method we show the existence of a complete family of axially symmetric solutions with a range of logarithmic growth rates which may be regarded as the analogue of the family of catenoids and hence called two end solutions nonexistence of two end solution with a small growth rate is also shown which differs from the theory of minimal surfaces

on de giorgi s conjecture in dimension n 9 a celebrated conjecture due to de giorgi states that any bounded solution of the equation u 1 u 2 u 0 in r n with ynu 0 must be such that its level setsfu g are all hyperplanes at least for dimension n 8 a counterexample for n 9 has long been believed to exist starting from a minimal graph which is not a hyperplane found by bombieri de giorgi and giusti in r n n 9 we prove that for any small 0 there is a bounded solution u y with yn u 0 which resembles tanh a t p 2 a where t t y denotes a choice of signed distance to the blown up minimal graph 1 this solution is a counterexample to de giorgi s conjecture for n 9