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

a liouville type theorem for schrodinger operators in this paper we prove a sufficient condition in terms of the behavior of a ground state of a symmetric critical operator p 1 such that a nonzero subsolution of a symmetric nonnegative operator p 0 is a ground state particularly if p j v j for j 0 1 are two nonnegative schrodinger operators defined on omega subseteq mathbb r d such that p 1 is critical in with a ground state the function psi nleq 0 is a subsolution of the equation p 0 u 0 in and satisfies psi leq c varphi in then p 0 is critical in and psi is its ground state in particular psi is up to a multiplicative constant the unique positive supersolution of the equation p 0 u 0 in similar results hold for general symmetric operators and also on riemannian manifolds

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

flatness of bernoulli jets level sets of minimizers of a possibly singular or degenerate elliptic variational problem related with fluid jets are shown to possess a harnack type inequality a few flatness and one dimensional results thence follow

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

moduli space theory for the allen cahn equation in the plane in this paper we study entire solutions of the allen cahn equation delta u f u 0 where f is an even bistable function we are particularly interested in the description of the moduli space of solutions which have some special structure at infinity the solutions we are interested in have their zero set asymptotic to 2k k 2 oriented affine half lines at infinity and along each of these affine half lines the solutions are asymptotic to the one dimensional heteroclinic solution such solutions are called multiple end solutions and their set is denoted by m 2k the main result of our paper states that if u in m 2k is nondegenerate then locally near u the set of solutions is a smooth manifold of dimension 2k this paper is part of a program whose aim is to classify all 2k ended solutions of the allen cahn equation in dimension 2 for k 2

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

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

gidas ni nirenberg results for finite difference equations estimates of approximate symmetry are positive solutions of finite difference boundary value problems hu f u in h u 0 on h as symmetric as the domain to answer this question we first show by examples that almost arbitrary non symmetric solutions can be constructed this is in striking difference to the continuous case where by the famous gidas ni nirenberg theorem b gidas wei ming ni l nirenberg symmetry and related problems via the maximum principle comm math phys 68 1979 209 243 positive solutions inherit the symmetry of the underlying domain then we prove approximate symmetry theorems for solutions on equidistantly meshed n dimensional cubes explicit estimates depending on the data are given which show that the solutions become more symmetric as the discretization gets finer the quality of the estimates depends on whether or not f 0 0 the one dimensional case stands out in two ways the proofs are elementary and the estimates for the defect of symmetry are o h compared to o 1 log h in the higher dimensional case

on variational characterization of four end solutions of the allen cahn equation in the plane abstract in this paper a novel variational method is developed to construct four end solutions in r 2 for the allen cahn equation four end solutions have been constructed by del pino kowalczyk pacard and wei when the angle of the ends is close to 2 or 0 using the lyapunov schmid reduction method and later by kowalczyk liu and pacard using a continuation method for general 0 2 by a special mountain pass argument in a restricted space namely a set of monotone paths of monotone functions a family of solutions in bounded domain is constructed with very good control of their nodal sets which then leads to a four end solution after sending the domains to r 2 in this way not only a family of four end solutions is constructed for any angle 0 2 the morse index of such a four end solution is also shown to be one

a new image segmentation method based on multifractal detrended moving average analysis abstract in order to segment and delineate some regions of interest in an image we propose a novel algorithm based on the multifractal detrended moving average analysis mf dma in this method the generalized hurst exponent h q is calculated for every pixel firstly and considered as the local feature of a surface and then a multifractal detrended moving average spectrum mf dms d h q is defined by the idea of box counting dimension method therefore we call the new image segmentation method mf dms based algorithm the performance of the mf dms based method is tested by two image segmentation experiments of rapeseed leaf image of potassium deficiency and magnesium deficiency under three cases namely backward 0 centered 0 5 and forward 1 with different q values the comparison experiments are conducted between the mf dms method and other two multifractal segmentation methods namely the popular mfs based and latest mf dfs based methods the results show that our mf dms based method is superior to the latter two methods the best segmentation result for the rapeseed leaf image of potassium deficiency and magnesium deficiency is from the same parameter combination of 0 5 and d h 10 when using the mf dms based method an interesting finding is that the d h 10 outperforms other parameters for both the mf dms based method with centered case and mf dfs based algorithms by comparing the multifractal nature between nutrient deficiency and non nutrient deficiency areas determined by the segmentation results an important finding is that the gray value s fluctuation in nutrient deficiency area is much severer than that in non nutrient deficiency area

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

hair triggered instability of radial steady states spread and extinction in semilinear heat equations we first study the initial value problem for a general semilinear heat equation we prove that every bounded nonconstant radial steady state is unstable if the spatial dimension is low n 10 or if the steady state is flat enough at infinity the solution of the heat equation either becomes unbounded as t approaches the lifespan or eventually stays above or below another bounded radial steady state depending on if the initial value is above or below the first steady state moreover the second steady state must be a constant if n 10 using this instability result we then prove that every nonconstant radial steady state of the generalized fisher equation is a hair trigger for two kinds of dynamical behavior extinction and spreading we also prove more criteria on initial values for these types of behavior similar results for a reaction diffusion system modeling an isothermal autocatalytic chemical reaction are also obtained

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

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

entire solutions of the allen cahn equation and complete embedded minimal surfaces of finite total curvature in r 3 we consider minimal surfaces m which are complete embedded and have finite total curvature in mathbb r 3 and bounded entire solutions with finite morse index of the allen cahn equation delta u f u 0 in mathbb r 3 here f w with w bi stable and balanced for instance w u frac 1 4 1 u 2 2 we assume that m has m ge 2 ends and additionally that m is non degenerate in the sense that its bounded jacobi fields are all originated from rigid motions this is known for instance for a catenoid and for the costa hoffman meeks surface of any genus we prove that for any small alpha gt 0 the allen cahn equation has a family of bounded solutions depending on m 1 parameters distinct from rigid motions whose level sets are embedded surfaces lying close to the blown up surface m alpha alpha 1 m with ends possibly diverging logarithmically from m alpha we prove that these solutions are l infty non degenerate up to rigid motions and find that their morse index coincides with the index of the minimal surface our construction suggests parallels of de giorgi conjecture for general bounded solutions of finite morse index

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

prime number theorems and holonomies for hyperbolic rational maps for a hyperbolic rational map f of degree at least two on the riemann sphere we obtain estimates for the number of primitive periodic orbits of f ordered by their multiplier and establish equidistribution of the associated holonomies both with power saving error terms

a geometric inequality in the heisenberg group and its applications to stable solutions of semilinear problems in the heisenberg group framework we obtain a geometric inequality for stable solutions of delta mathbb h u f u in a domain omega subseteq mathbb h more precisely if we denote the horizontal intrinsic hessian by hu the mean curvature of a level set by h its imaginary curvature by p the intrinsic normal by and the unit tangent by we have that int limits omega nabla mathbb h phi 2 nabla mathbb h u 2 quad geq int limits omega cap nabla mathbb h u ne0 left hu 2 langle hu 2 nu nu rangle mathbb h 2 left tyuxu txuyu right right phi 2 quad int limits omega cap nabla mathbb h u ne0 nabla mathbb h u 2 big h 2 big p frac langle huv nu rangle mathbb h nabla mathbb h u big 2 2 langle t nu v rangle mathbb h big phi 2 for any phi in c infty 0 omega stable solutions in the entire mathbb h satisfying a suitably weighted energy growth and such that langle t nu v rangle mathbb h ge0 are then shown to have level sets with vanishing mean curvature

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

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

multiple end solutions to the allen cahn equation in r2 we construct a new class of entire solutions for the allen cahn equation u 1 u2 u 0 in r2 c given k 1 we find a family of solutions whose zero level sets are away from a compact set asymptotic to 2k straight lines which we call the ends these solutions have the property that there exist 0 1 2k 0 2 such that limr u rei 1 j uniformly in on compact subsets of j j 1 for j 0 2k 1