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

phase transition with the line tension effect we make the connection between the geometric model for capillarity with line tension and the cahn hilliard model of two phase fluids to this aim we consider the energies f eps u eps int o du 2 1 over eps int o w u lambda int bd o v u where u is a scalar density function and w and v are double well potentials we show that the behaviour of f e in the limit e 0 and depends on the limit of e log if this limit is finite and strictly positive then the singular limit of the energies f e leads to a coupled problem of bulk and surface phase transitions and under certain assumptions agrees with the relaxation of the capillary energy with line tension these results were announced in abs1 and abs2

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

singular perturbation of domains and the semilinear elliptic equation ii on considere un domaine a perturbation singuliere d 1 d 2 q avec un petit parametre 0 et on etudie le comportement des solutions et leur structure du probleme aux valeurs limites elliptique semilineaire pour quand 0 est petit v f v 0 dans v v 0 sur ou f est une fonction lisse a valeur reelle non lineaire

interfaces in some elliptic nonlinear boundary value problems on riemannian manifolds necessary condition and location we show that the equal area condition is a necessary hypothesis for the existence of layered solutions to a nonlinear boundary value problem depending on a positive real parameter on a riemannian manifold with or without boundary as an application we give examples where the results can be used in order to determine in advance the location of possible limiting interface as the parameter goes to zero

parabolic evolution equations and nonlinear boundary conditions on etudie le probleme aux valeurs limites et initiales u 7b a t u f t u 7b b t u g t u 0 t t u 0 u 0 qui peut etre considere comme une contrepartie abstraite a des problemes paraboliques semilineaires

bifurcation of stable equilibria under nonlinear flux boundary condition with null average weight abstract the bifurcation and stability structures of equilibria of a parabolic problem motivated by a population genetics model are completely described in a previous work the authors considered weights having nonzero average and drew the bifurcation and stability diagrams of equilibria herein that study is completed by considering weights with null average over the boundary of the domain and other approach is employed to overcome a degeneration which occurs in such case for each value of the parameter we prove existence of an unique nonconstant equilibrium solution it belongs to a globally parametrized bifurcation curve constructed using a lyapunov schmidt type reduction combined with the morse lemma that solution is showed to be a global minimizer of the corresponding energy functional which attracts all nontrivial semi orbits the behavior of its trace when the parameter is large is established and complete diagrams containing all bifurcation and stability information are also provided

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

model for line tension in three phase equilibrium we propose a model free energy functional of two order parameters with which to calculate the interfacial and line tensions in three phase equilibrium the euler lagrange equations for the free energy minimum are solved exactly yielding the spatial variation of the order parameters analytically in terms of a parameter b2 in the model the three interfacial tensions in dimensionless form are 12 1 b2 12 1 b2 and 2 when b2 3 the three phases play symmetrical roles and the line tension again in the appropriate units is calculated to be 6 2 3 0 755 a wetting transition where the sum of two of the interfacial tensions becomes equal to the third occurs as b2 1 a quantity that approximates the line tension is found to vanish proportionally to the first power of the vanishing contact angle as the wetting transition is approached

shortcut sets for the locus of plane euclidean networks abstract we study the problem of augmenting the locus n l of a plane euclidean network n by inserting iteratively a finite set of segments called shortcut set while reducing the diameter of the locus of the resulting network there are two main differences with the classical augmentation problems the endpoints of the segments are allowed to be points of n l as well as points of the previously inserted segments instead of only vertices of n and the notion of diameter is adapted to the fact that we deal with n l instead of n this increases enormously the hardness of the problem but also its possible practical applications to network design among other results we characterize the existence of shortcut sets compute them in polynomial time and analyze the role of the convex hull of n l when inserting a shortcut set our main results prove that while the problem of minimizing the size of a shortcut set is np hard one can always determine in polynomial time whether inserting only one segment suffices to reduce the diameter

a nonlinear parabolic equation with varying domain abstract this document considers a boundary value problem where the basic problem is to understand how new equilibrium solutions may appear for a given nonlinear function f as the domain omega is changed these new solutions being bifurcations from known equilibrium solutions for a given cubic function f an intuitive discussion of this phenomenon was given by hale indicating how stable non constant equilibria appear as secondary rather than primary bifurcations additional keywords continuity perturbations reprints

layered stable equilibria of a reaction diffusion equation with nonlinear neumann boundary condition abstract in this work we investigate the existence and asymptotic profile of a family of layered stable stationary solutions to the scalar equation u t e 2 u f u in a smooth bounded domain r 3 under the boundary condition e u e g u it is assumed that has a cross section which locally minimizes area and lim e 0 e ln e with 0 and e 1 when 0 the functions f and g are of bistable type and do not necessarily have the same zeros what makes the asymptotic geometric profile of the solutions on the boundary to be different from the one in the interior

improving the stretch factor of a geometric network by edge augmentation given a euclidean graph g in mathbb r d with n vertices and m edges we consider the problem of adding an edge to g such that the stretch factor of the resulting graph is minimized currently the fastest algorithm for computing the stretch factor of a graph with positive edge weights runs in cal o nm n 2 log n time resulting in a trivial cal o n 3m n 4 log n time algorithm for computing the optimal edge first we show that a simple modification yields the optimal solution in cal o n 4 time using cal o n 2 space to reduce the running time we consider several approximation algorithms

euclidean chains and their shortcuts a euclidean graph is a straight line embedding of a graph in the plane such that there is no crossing between any pair of edges and the length of an edge is the euclidean distance between its two endpoints a euclidean chain c x 1 x 2 x n is a planar straight line graph with vertex set x 1 x 2 x n and edge set x ix i 1 1 i n 1 given a euclidean chain c in the plane we study the problem of finding a pair of points on c such that the new euclidean graph obtained from c by adding a straight line segment called a shortcut between this pair of points has the minimum diameter we also study the ratio between the diameter of the new graph and the length of c we give necessary and sufficient conditions for optimal shortcuts we present an approximation algorithm for computing the optimal shortcuts of chains in addition we introduce two types of chains a strongly monotonic chain and a simple chain we provide properties for these two types of chains and their shortcuts

bifurcation of stable equilibria and nonlinear flux boundary condition with indefinite weight abstract we study bifurcation and stability of positive equilibria of a parabolic problem under a nonlinear neumann boundary condition having a parameter and an indefinite weight the main motivation is the selection migration problem involving two alleles and no gene flux acrossing the boundary introduced by fisher and fleming and henry s approach to the problem local and global structures of the set of equilibria are given while the stability of constant equilibria is analyzed the exponential stability of the unique bifurcating nonconstant equilibrium solution is established diagrams exhibiting the bifurcation and stability structures are also furnished moreover the asymptotic behavior of such solutions on the boundary of the domain as the positive parameter goes to infinity is also provided the results are obtained via classical tools like the implicit function theorem bifurcation from a simple eigenvalue theorem and the exchange of stability principle in a combination with variational and dynamical arguments

analysis of a viscoelastic spring mass model in this paper we consider a linear wave equation with strong damping and dynamical boundary conditions as an alternative model for the classical spring mass damper ode our purpose is to compare analytically these two approaches to the same physical system we take a functional analysis point of view based on semigroup theory spectral perturbation analysis and dominant eigenvalues

bistable boundary reactions in two dimensions in a bounded domain omega subset mathbb r 2 with smooth boundary we consider the problem delta u 0 quad rm in omega qquad frac partial u partial nu frac1 varepsilon f u quad rm on partial omega where is the unit normal exterior vector e 0 is a small parameter and f is a bistable nonlinearity such as f u sin u or f u 1 u2 u we construct solutions that develop multiple transitions from 1 to 1 and vice versa along a connected component of the boundary we also construct an explicit solution when is a disk and f u sin u

exponential representation of jordanian matrix quantum group glh 2 an exponential representation of the jordanian matrix quantum group glh 2 is constructed in explicit form by the generators of the classical lie algebra sl 2 c

semigroups and nonlinear evolution equations abstract in the main part of this paper we survey some recent results on a semigroup approach to parabolic equations under nonlinear boundary conditions in particular we give a detailed motivation and a heuristic derivation of a generalized variation of constant formula which is the basis for the whole method in the appendix we prove some new results about semigroups in interpolation and extrapolation spaces

dynamics in dumbbell domains i continuity of the set of equilibria we analyze the dynamics of a reaction diffusion equation with homogeneous neumann boundary conditions in a dumbbell domain we provide an appropriate functional setting to treat this problem and as a first step we show in this paper the continuity of the set of equilibria and of its linear unstable manifolds

patterns in parabolic problems with nonlinear boundary conditions we obtain existence of asymptotically stable nonconstant equilibrium solutions for semilinear parabolic equations with nonlinear boundary conditions on small domains connected by thin channels we prove the convergence of eigenvalues and eigenfunctions of the laplace operator in such domains this information is used to show that the asymptotic dynamics of the heat equation in this domain is equivalent to the asymptotic dynamics of a system of two ordinary differential equations diffusively weakly coupled the main tools employed are the invariant manifold theory and a uniform trace theorem

some remarks on the gauss decomposition for quantum group glq n with application to q bosonization the gauss decomposition of quantum group glq n is defined purely algebraically the entries of the upper triangular matrix commutes with those of the lower triangular one the quantum determinant is the product of the diagonal matrix elements this decomposition gives rise to a new bosonization of the glq n in terms of 2n q oscillators

limiting behavior for strongly damped nonlinear wave equations in this paper we investigate both the existence and the limiting behavior for the equation utt aut au t u ut where a is a sectorial operator is periodic in t and satisfies certain regularity and growth assumptions in most results on limiting behavior we will assume a has compact resolvent we consider the equation as an abstract ode defined on a paired space x x 0 1 with regard to the limiting behavior one of our principal results will be to show that if there is a bounded set in one of the spaces considered for which all points or trajectories enter into and remain then there is a set j consisting of very smooth functions defined on all of the spaces considered which is the maximum compact invariant set uniformly asymptotically stable connected and having very strong attractivity properties in all these spaces we will often show it attracts all points in a bounded set uniformly we will give a few sharper results for the case where a the work is motivated by recent papers of webb and fitzgibbon and applies techniques found in recent papers by the author

necessity of internal and boundary bulk balance law for existence of interfaces for an elliptic system with nonlinear boundary condition abstract for a system of stationary solutions to a reaction diffusion equations with small diffusion coefficient and nonlinear flux boundary condition we prove that the bulk balance law not only on the domain but on its boundary as well is a necessary condition for formation of internal and boundary layers

q difference intertwining operators for uq sl n general setting and the case n 3 we construct representations pi r of the quantum algebra uq sl n labelled by n 1 complex numbers ri and acting in the space of formal power series of n n 1 2 non commuting variables these variables generate a flag manifold of the matrix quantum group slq n which is dual to uq sl n the conditions for reducibility of pi r and the procedure for the construction of the q difference intertwining operators are given the representations and q difference intertwining operators are given in the most explicit form for n 3

a model of porous catalyst accounting for incipiently non isothermal effects abstract an approximate model accounting for incipiently non isothermal effects is derived from a well known model of porous catalyst for appropriate realistic limiting values of the parameters in this limit the original model is a singularly perturbed m d reaction diffusion system and the approximate model is given by the m d heat equation with nonlinear boundary condition coupled with infinitely many ifm 2 1 d semilinear parabolic equations one for each point of the boundary of the spatial domain some limiting cases are still considered in the approximate model that lead to further simplifications

lichnerowicz type equations with sign changing nonlinearities on complete manifolds with boundary abstract we prove an existence theorem for positive solutions to lichnerowicz type equations on complete manifolds with boundary m m and nonlinear neumann conditions this kind of nonlinear problems arise quite naturally in the study of solutions for the einstein scalar field equations of general relativity in the framework of the so called conformal method