fractional laplacian phase transitions and boundary reactions a geometric inequality and a symmetry result we deal with symmetry properties for solutions of nonlocal equations of the type sv f v in rn where s 0 1 and the operator s is the so called fractional laplacian the study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation div x u 0on rn 0 x ux f u on rn 0 where 1 1 y rn x 0 and u u y x this equation is related to the fractional laplacian since the dirichlet to neumann operator u r n 1 x ux r n 1 is 1 2 more generally we study the so called boundary reaction equations given by div x u g x u 0on rn 0 x ux f u on rn 0 under some natural assumptions on the diffusion coefficient and on the nonlinearities f and g we prove a geometric formula of poincare type for stable solutions from which we derive a symmetry result in the spirit of a conjecture of de giorgi

serrin s overdetermined problem and constant mean curvature surfaces for all n 9 we find smooth entire epigraphs in rn namely smooth domains of the form x rn xn f x1 xn 1 which are not half spaces and in which a problem of the form u f u 0 in has a positive bounded solution with 0 dirichlet boundary data and constant neumann boundary data on this answers negatively for large dimensions a question by berestycki caffarelli and nirenberg in 1971 serrin proved that a bounded domain where such an overdetermined problem is solvable must be a ball in analogy to a famous result by alexandrov that states that an embedded compact surface with constant mean curvature cmc in euclidean space must be a sphere in lower dimensions we succeed in providing examples for domains whose boundary is close to large dilations of a given cmc surface where serrin s overdetermined problem is solvable

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

layer solutions in a half space for boundary reactions r n 1 are denoted by y y1 yn 1 our main goal is to study bounded solutions of 1 1 that are monotone increasing say from 1 to 1 in one of the y variables we call them layer solutions of 1 1 and we study their existence uniqueness symmetry and variational properties as well as their asymptotic behavior the interest in such increasing solutions comes from some models of boundary phase transitions when the nonlinearity f is given by f u sin cu for some constant c problem 1 1 in a half plane is called the peierls nabarro problem and it appears as a model of dislocations in crystals see 21 36 the peierls nabarro problem is also central to the analysis of boundary vortices in the paper 28 which studies a model for soft thin films in micromagnetism recently derived by kohn and slastikov 26 see also 27 our main result theorem 1 2 characterizes the nonlinearities f for which there exists a layer solution of 1 1 in dimension n 2 we prove that the necessary and sufficient condition is that the potential g defined by g f has two and only two absolute minima in the interval 1 1 located at 1 under the additional hypothesis g 1 0 we also establish the uniqueness of a layer solution up to translations in the y variable the proofs of both the necessity and the sufficiency of the condition on g for existence use new ingredients which we develop in this article a first one is a nonlocal estimate as well as a conserved or hamiltonian quantity satisfied by every layer solution in dimension 2 see theorem 1 3 the estimate can be seen as

stable and finite morse index solutions on ⁿ or on bounded domains with small diffusion in this paper we study bounded solutions of u f u on r n where n 2 and sometimes n 3 and show that for most f s the weakly stable and finite morse index solutions are quite simple we then use this to obtain a very good understanding of the stable and bounded morse index solutions of e 2 u f u on with dirichlet or neumann boundary conditions for small e

clifford tori and the singularly perturbed cahn hilliard equation abstract in this paper we construct entire solutions u e to the cahn hilliard equation e 2 e 2 u w u w u e 2 u w u e 4 e 1 u e under the volume constraint r 3 1 u e 2 d x 8 2 2 c e with c e 1 as e 0 whose nodal set approaches the clifford torus that is the torus with radii of ratio 1 2 embedded in r 3 as e 0 it is crucial that the clifford torus is a willmore hypersurface and it is non degenerate up to conformal transformations the proof is based on the lyapunov schmidt reduction and on careful geometric expansions of the laplacian

symmetry of some entire solutions to the allen cahn equation in two dimensions abstract in this paper we prove even symmetry and monotonicity of certain solutions of allen cahn equation in a half plane we also show that entire solutions with finite morse index and four ends must be evenly symmetric with respect to two orthogonal axes a classification scheme of general entire solutions with finite morse index is also presented using energy quantization

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

on stable solutions for boundary reactions a de giorgi type result in dimension 4 1 we prove that every bounded stable solution of begin aligned delta 1 2 u f u 0 qquad text in mathbb r 3 end aligned is a 1d profile i e u x phi e cdot x for some e in mathbb s 2 where phi mathbb r rightarrow mathbb r is a nondecreasing bounded stable solution in dimension one equivalently stable critical points of boundary reaction problems in mathbb r d 1 mathbb r d 1 cap x d 1 ge 0 of the form begin aligned int x d 1 ge 0 frac 1 2 nabla u 2 dx dx d 1 int x d 1 0 f u dx end aligned are 1d when d 3 these equations have been studied since the 1940 s in crystal dislocations also as it happens for the allen cahn equation the associated energies enjoy a gamma convergence result to the perimeter functional in particular when f u u 3 u or equivalently when f u frac 1 4 1 u 2 2 our result implies the analogue of the de giorgi conjecture for the half laplacian in dimension 4 namely that monotone solutions are 1d note that our result is a pde version of the fact that stable embedded minimal surfaces in mathbb r 3 are planes it is interesting to observe that the corresponding statement about stable solutions to the allen cahn equation namely when the half laplacian is replaced by the classical laplacian is still unknown for d 3

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

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

some monotonicity results for minimizers in the calculus of variations abstract we obtain monotonicity properties for minima and stable solutions of general energy functionals of the type f u u x d x under the assumption that a certain integral grows at most quadratically at infinity as a consequence we obtain several rigidity results of global solutions in low dimensions

the gibbs thompson relation within the gradient theory of phase transitions this paper discusses the asymptotic behavior as 0 of the chemical potentials associated with solutions of variational problems within the van der waals cahn hilliard theory of phase transitions in a fluid with free energy per unit volume given by 2 2 w where is the density the main result is that is asymptotically equal to e d o with e the interfacial energy per unit surface area of the interface between phases the constant sum of principal curvatures of the interface and d the density jump across the interface this result is in agreement with a formula conjectured by m gurtin and corresponds to the gibbs thompson relation for surface tension proved by g caginalp within the context of the phase field model of free boundaries arising from phase transitions

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

a general halfspace theorem for constant mean curvature surfaces in this paper we prove a general halfspace theorem for constant mean curvature surfaces under certain hypotheses we prove that in an ambient space m 3 any constant mean curvature h 0 surface on one side of a constant mean curvature h 0 surface sigma 0 is an equidistant surface to sigma 0 the main hypotheses of the theorem are that sigma 0 is parabolic and the mean curvature of the equidistant surfaces to sigma 0 evolves in a certain way

regularity results and harnack inequalities for minimizers and solutions of nonlocal problems a unified approach via fractional de giorgi classes abstract we study energy functionals obtained by adding a possibly discontinuous potential to an interaction term modeled upon a gagliardo type fractional seminorm we prove that minimizers of such non differentiable functionals are locally bounded holder continuous and that they satisfy a suitable harnack inequality hence we provide an extension of celebrated results of m giaquinta and e giusti to the nonlocal setting to do this we introduce a particular class of fractional sobolev functions reminiscent of that considered by e de giorgi in his seminal paper of 1957 the flexibility of these classes allows us to also establish regularity of solutions to rather general nonlinear integral equations

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

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

nonlinear equations for fractional laplacians ii existence uniqueness and qualitative properties of solutions this paper which is the follow up to part i concerns the equation delta s v g v 0 in mathbb r n with s in 0 1 where delta s stands for the fractional laplacian the infinitesimal generator of a l evy process when n 1 we prove that there exists a layer solution of the equation i e an increasing solution with limits pm 1 at pm infty if and only if the potential g has only two absolute minima in 1 1 located at pm 1 and satisfying g 1 g 1 0 under the additional hypothesis g 1 0 and g 1 0 we also establish its uniqueness and asymptotic behavior at infinity furthermore we provide with a concrete almost explicit example of layer solution for n geq 1 we prove some results related to the one dimensional symmetry of certain solutions in the spirit of a well known conjecture of de giorgi for the standard laplacian

eulerian solutions of o σnv for the stochastic transport problem for conservative tracers coupled with o σ4ƒ solutions for the flow problem in an infinite domain an o vn eulerian stochastic theory cushman and hu 1997 for conservative transport is coupled with an o f4 eulerian solution to the flow problem the theory provides a self consistent recursive solution to the closure problem associated with eulerian methods the stochastic concentration is given to arbitrary order in v the variance of fluctuating velocity the balance law for mean concentration is not required in the analysis closed forms for mean concentration are given up to o v4 the mean concentration is transformed to wave vector and frequency domain via fast fourier transform to calculate numerically the mean concentration as well as its spatial moments the results show that the first order solution in v2 is equivalent to the nonlocal theory ofdeng et al 1993 second order corrections to flow and transport equations slightly decrease the second longitudinal moment but significantly increase the second transverse moment which is consistent with hsu et al s 1996 results the influence of second order corrections to skewness is not clear the second longitudinal moment obtained from the second order correction agrees with the monte carlo result but second order results for the second transverse moment and skewness significantly differ from those given in monte carlo simulations coupling the transport correction models with the velocity covariance generated through monte carlo simulation gives second transverse spatial moments that are very close to monte carlo simulations which suggests that the correction to flow is more important than the correction to transport the results also bring into question the accuracy of monte carlo simulations for flow

some symmetry results and liouville type theorems for solutions to semilinear equations abstract we study the one dimensional symmetry and the monotonicity of positive and bounded solutions to semilinear elliptic equations in euclidean half spaces we also prove some liouville type theorems for stable and finite morse index entire solutions

a geometric inequality and a symmetry result for elliptic systems involving the fractional laplacian we study the symmetry properties for solutions of elliptic systems of the type s1u f1 u v s2v f2 u v where f cloc1 1 r2 s1 s2 0 1 and the operator s is the so called fractional laplacian we obtain some poincare type formulas for the harmonic extension in the half space that we use to prove a symmetry result both for stable and for monotone solutions

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

a liouville theorem for the euler equations in the plane this paper is concerned with qualitative properties of bounded steady flows of an ideal incompressible fluid with no stagnation point in the two dimensional plane mathbb r 2 we show that any such flow is a shear flow that is it is parallel to some constant vector the proof of this liouville type result is firstly based on the study of the geometric properties of the level curves of the stream function and secondly on the derivation of some estimates on the at most logarithmic growth of the argument of the flow in large balls these estimates lead to the conclusion that the streamlines of the flow are all parallel lines

characterization of solutions having finite morse index for some nonlinear pde with supercritical growth abstract in this paper we study solutions of the equation u u p in r n where n 2 n 2 p p c n see 3 for the definition of p c n we first classify all solutions not necessarily bounded having finite morse index for r n or r n with dirichlet boundary condition when is a regular bounded domain of r n for a family of solutions with dirichlet boundary condition u n we prove that u n l is bounded if and only if the morse index sequence i u n of u n is bounded finally we prove the same result for other nonlinearities f x u which have similar growth respectively at

on phase separation models asymptotics and qualitative properties in this paper we study bound state solutions of a class of two component nonlinear elliptic systems with a large parameter tending to infinity the large parameter giving strong intercomponent repulsion induces phase separation and forms segregated nodal domains divided by an interface to obtain the profile of bound state solutions near the interface we prove the uniform lipschitz continuity of bound state solutions when the spatial dimension is n 1 furthermore we show that the limiting nonlinear elliptic system that arises has unbounded solutions with symmetry and monotonicity these unbounded solutions are useful for rigorously deriving the asymptotic expansion of the minimizing energy which is consistent with the hypothesis of du and zhang discontin dynam sys 2012 when the spatial dimension is n 2 we establish the de giorgi type conjecture for the blow up nonlinear elliptic system under suitable conditions at infinity on bound state solutions these results naturally lead us to formulate de giorgi type conjectures for these types of systems in higher dimensions

symmetry of traveling wave solutions to the allen cahn equation in mathbb r 2 in this paper we prove even symmetry of monotone traveling wave solutions to the balanced allen cahn equation in the entire plane related results for the unbalanced allen cahn equation are also discussed

optimal profiles in a phase transition model with a saturating flux abstract it is well known that for the allen cahn equation the minimizing transition in an infinite cylinder r is one dimensional and unique up to a translation in the first variable we analyze in this paper the existence and symmetry of optimal profiles for transitions in a similar phase separation model with a saturating flux this amounts to consider transitions in the space of bv functions as we consider the area integral instead of the dirichlet energy to penalize the creation of wild interfaces

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

an extension problem for sums of fractional laplacians and 1 d symmetry of phase transitions we study nonlinear elliptic equations for operators corresponding to non stable levy diffusions we include a sum of fractional laplacians of different orders such operators are infinitesimal generators of non stable i e non self similar levy processes we establish the regularity of solutions as well as sharp energy estimates as a consequence we prove a 1 d symmetry result for monotone solutions to allen cahn type equations with a non stable levy diffusion these operators may still be realized as local operators using a system of pdes in the spirit of the extension problem of caffarelli and silvestre

a gradient bound and a liouville theorem for nonlinear poisson equations soit f c 2 r une fonction non negative et u c 3 r n une solution dans tout r n de l equation u f u ou f f est la derivee 1ere de f si u est borne sur r n et s il existe x 0 r n tel que f u x 0 0 alors u est constante