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

on a free boundary problem for minimal surfaces forc 4 embedded manifoldss 3 which are differmorphic to the standard sphere in 3 the existence of non constant minimal surfaces bounded bys and intersectings orthogonally along their boundaries is deduced

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

an implicit midpoint difference scheme for the fractional ginzburg landau equation this paper proposes and analyzes an efficient difference scheme for the nonlinear complex ginzburg landau equation involving fractional laplacian the scheme is based on the implicit midpoint rule for the temporal discretization and a weighted and shifted grunwald difference operator for the spatial fractional laplacian by virtue of a careful analysis of the difference operator some useful inequalities with respect to suitable fractional sobolev norms are established then the numerical solution is shown to be bounded and convergent in the l h 2 norm with the optimal order o 2 h 2 with time step and mesh size h the a priori bound as well as the convergence order holds unconditionally in the sense that no restriction on the time step in terms of the mesh size h needs to be assumed numerical tests are performed to validate the theoretical results and effectiveness of the scheme

critical exponents for a nonlinear diffusion system abstract this paper deals with critical exponents for a nonlinear diffusion system with inner absorptions and coupled nonlinear boundary flux a precise analysis of the interactions among the multi nonlinearities in the system is given to determine the critical exponents which are simply described via the signs of a pair of parameters 1 2 solving the characteristic algebraic system introduced the solutions are non global for large initial data if 1 1 or 1 2 0 and global if 1 1 1 2 0 while in the critical case of 1 1 1 2 0 0 the solutions can be both global and non global depending on the geometry and the size of the domain

diffusivity versus absorption through the boundary for all 0 1 then we do expect to find that any solution becomes unbounded in finite time see 13 15 63 it is our goal to prove an analogous result for m f 1 for a 1 one can find that if v is a positive solution of

energy minimisers of prescribed winding number in an s1 valued nonlocal allen cahn type model abstract we study a variational model for transition layers with an underlying functional that combines an allen cahn type structure with an additional nonlocal interaction term a transition layer is represented by a map from r to s 1 thus it has a topological invariant in the form of a winding number and we study minimisers subject to a prescribed winding number as shown in our previous paper 14 the nonlocal term gives rise to solutions that would not be present for a functional including only the local allen cahn terms we complete the picture here by proving existence of minimisers in all cases where it has been conjectured we also prove non existence in some other cases finally in addition to existence we prove a result for the structure of minimizers

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

global existence and nonexistence for semilinear parabolic systems with nonlinear boundary conditions on etudie des problemes aux valeurs limites et initiales paraboliques semilineaires de la forme t u j a jk k u a 0 u f u dans 0 a jk j k u bu 1 u g u sur 0 u 0 u 0 sur ou est un domaine borne de r n n 1 de classe c 2

weak solutions for the time fractional harmonic map flow abstract we consider the time fractional harmonic map flow on a smooth bounded domain and prove the existence of weak solutions

on regularity theory for n p harmonic maps into manifolds abstract in this paper we continue the investigation started in the paper da lio and schikorra 2014 of the regularity of the so called weak n p harmonic maps in the critical case these are critical points of the following nonlocal energy l s u r n s 2 u x p d x where u h s p r n n and n r n is a closed k dimensional smooth manifold and s n p we prove holder continuity for such critical points for p 2 for p 2 we obtain the same under an additional lorentz space assumption the regularity theory is in the two cases based on regularity results for nonlocal schrodinger systems with an antisymmetric potential

dynamics of degenerate quasilinear reaction diffusion systems with nonnegative initial functions abstract this paper is concerned with a system of quasilinear reaction diffusion equations with density dependent diffusion coefficients and mixed quasimonotone reaction functions the equations are allowed to be degenerate and the boundary conditions are of the nonlinear type the main goals are to prove the existence and uniqueness of the weak solution between a pair of coupled upper and lower solutions show that the weak solution evolves into the classical solution and analyze the asymptotic behavior of the solution using quasi solutions of the steady state system the general results are applied to a degenerate lotka volterra competition model conditions are given for the solution to exist globally to evolve into the classical solution and to be attracted into a sector formed by quasi solutions of the elliptic system especially for the neumann problem we give a simple condition for the solution to converge to a unique constant steady state solution which is a global attractor

existence and dynamics of quasilinear parabolic systems with time delays abstract this paper is concerned with a coupled system of quasilinear parabolic equations where the effect of time delays is taken into consideration in the reaction functions of the system the partial differential operators in the system may be degenerate and the reaction functions possess some mixed quasimonotone property including quasimonotone nondecreasing functions the aim of the paper is to show the existence and uniqueness of a global solution to the parabolic system the existence of positive quasisolutions or maximal minimal solutions of the corresponding elliptic system and the asymptotic behavior of the solution of the parabolic system in relation to the quasisolutions or maximal minimal solutions of the elliptic system applications are given to three reaction diffusion models arising from mathematical biology and ecology where the diffusion coefficients are density dependent and are degenerate this degenerate density dependent diffusion leads to some interesting distinct asymptotic behavior of the time dependent solution when compared with density independent diffusion

existence and partial regularity results for the heat flow for harmonic maps on considere le probleme d evolution t u m u n u 0 dans m r u 0 u 0 sur m ou m et n sont des varietes de riemann lisses compactes et u 0 l application harmonique de m dans n on demontre l existence d une solution globale faible

the method of upper and lower solutions for diffraction problems of quasilinear elliptic reaction diffusion systems abstract the aim of this paper is to show the existence of solutions of the n dimensional diffraction problem for weakly coupled quasilinear elliptic reaction diffusion system the coefficients of the equations under consideration are allowed to be discontinuous we extend the method of upper and lower solutions for reaction diffusion equations with continuous coefficients to the elliptic diffraction problem an application of these results is given to the steady state problem of lotka volterra cooperation model with two cooperating species

asymptotic stability of monotone increasing traveling wave solutions for viscous compressible fluid equations with capillarity term abstract asymptotic stability of monotone increasing traveling wave solutions is investigated for a class of viscous compressible fluid equations with capillarity term by using the theory of planar dynamical systems the existence has been proved for the monotone increasing traveling wave solution moreover through building uniformly a priori energy estimate for the perturbation of the traveling wave solution we prove that the monotone increasing traveling wave solution is asymptotically stable in h 2 h 1

fractional div curl quantities and applications to nonlocal geometric equations abstract we investigate a fractional notion of gradient and divergence operator we generalize the div curl estimate by coifman lions meyer semmes to fractional div curl quantities obtaining in particular a nonlocal version of wente s lemma we demonstrate how these quantities appear naturally in nonlocal geometric equations which can be used to obtain a theory for fractional harmonic maps analogous to the local theory firstly regarding fractional harmonic maps into spheres we obtain a conservation law analogous to shatah s conservation law and give a new regularity proof analogous to helein s for harmonic maps into spheres secondly we prove regularity for solutions to critical systems with nonlocal antisymmetric potentials on the right hand side since the half harmonic map equation into general target manifolds has this form as a corollary we obtain a new proof of the regularity of half harmonic maps into general target manifolds following closely riviere s celebrated argument in the local case lastly the fractional div curl quantities provide also a new simpler proof for holder continuity of w s n s harmonic maps into spheres and we extend this to an argument for w s n s harmonic maps into homogeneous targets this is an analogue of strzelecki s and toro wang s proof for n harmonic maps into spheres and homogeneous target manifolds respectively

fractional ginzburg landau equation for fractal media abstract we derive the fractional generalization of the ginzburg landau equation from the variational euler lagrange equation for fractal media to describe fractal media we use the fractional integrals considered as approximations of integrals on fractals some simple solutions of the ginzburg landau equation for fractal media are considered and different forms of the fractional ginzburg landau equation or nonlinear schrodinger equation with fractional derivatives are presented the agrawal variational principle and its generalization have been applied

asymptotics for the fractional allen cahn equation and stationary nonlocal minimal surfaces this article is mainly devoted to the asymptotic analysis of a fractional version of the elliptic allen cahn equation in a bounded domain omega subseteq mathbb r n with or without a source term in the right hand side of the equation commonly called chemical potential in contrast to the usual allen cahn equation the laplace operator is here replaced by the fractional laplacian delta s with s in 0 1 2 as defined in fourier space in the singular limit varepsilon to 0 we show that arbitrary solutions with uniformly bounded energy converge both in the energetic and geometric sense to surfaces of prescribed nonlocal mean curvature in whenever the chemical potential remains bounded in suitable sobolev spaces with no chemical potential the notion of surface of prescribed nonlocal mean curvature reduces to the stationary version of the nonlocal minimal surfaces introduced by caffarelli et al commun pure appl math 63 1111 1144 2010 under the same sobolev regularity assumption on the chemical potential we also prove that surfaces of prescribed nonlocal mean curvature have a minkowski codimension equal to one and that the associated sets have a locally finite fractional 2s prime perimeter in for every s prime in 0 1 2

positive solutions of quasilinear parabolic systems with dirichlet boundary condition abstract coupled systems for a class of quasilinear parabolic equations and the corresponding elliptic systems including systems of parabolic and ordinary differential equations are investigated the aim of this paper is to show the existence uniqueness and asymptotic behavior of time dependent solutions also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system the elliptic operators in both systems are allowed to be degenerate in the sense that the density dependent diffusion coefficients d i u i may have the property d i 0 0 for some or all i 1 n and the boundary condition is u i 0 using the method of upper and lower solutions we show that a unique global classical time dependent solution exists and converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions and if the maximal and minimal solutions coincide then the steady state solution is unique and the time dependent solution converges to the unique solution applications of these results are given to three model problems including a scalar polynomial growth problem a coupled system of polynomial growth problem and a two component competition model in ecology

partial regularity for stationary harmonic maps into spheres in an interesting recent paper 12 f hi lein has shown that any weakly harmonic mapping from a two dimensional surface into a sphere is smooth i present here a kind of generalization to higher dimensions asserting in effect that a stationary harmonic mapping from an open subset of nn n 3 into a sphere is smooth except possibly for a closed singular set of n 2 dimensional hausdorff measure zero my proof crucially depends upon several of h l in s observations as streamlined by p l lions to state the result precisely let us suppose that m n 2 u is a smooth open subset of nn and s m 1 denotes the unit sphere in nm a function u in the sobolev space hi u rm u u 1 urn belongs to h i u s ml provided t u 1 a e in u

an efficient split step quasi compact finite difference method for the nonlinear fractional ginzburg landau equations abstract in this paper we propose a split step quasi compact finite difference method to solve the nonlinear fractional ginzburg landau equations both in one and two dimensions the original equations are split into linear and nonlinear subproblems the riesz space fractional derivative is approximated by a fourth order fractional quasi compact method furthermore an alternating direction implicit scheme is constructed for the two dimensional linear subproblem the unconditional stability and convergence of the schemes are proved rigorously in the linear case numerical experiments are performed to confirm our theoretical findings and the efficiency of the proposed method

quasilinear reaction diffusion systems with nonlinear boundary conditions this paper deals with the existence and nonexistence of global positive solutions of quasilinear reaction diffusion systems with nonlinear boundary conditions necessary and sufficient conditions on the global existence of all positive solutions are obtained

on a fractional ginzburg landau equation and 1 2 harmonic maps into spheres this paper is devoted to the asymptotic analysis of a fractional version of the ginzburg landau equation in bounded domains where the laplacian is replaced by an integro differential operator related to the square root laplacian as defined in fourier space in the singular ginzburg landau limit we show that solutions with uniformly bounded energy converge weakly to sphere valued 1 2 harmonic maps i e the fractional analogues of the usual harmonic maps in addition the convergence holds in smooth functions spaces away from a n 1 rectifiable closed set of finite n 1 hausdorff measure the proof relies on the representation of the square root laplacian as a dirichlet to neumann operator in one more dimension and on the analysis of a boundary version of the ginzburg landau equation besides the analysis of the fractional ginzburg landau equation we also give a general partial regularity result for stationary 1 2 harmonic maps in arbitrary dimension

doubly nonlinear degenerate parabolic systems with coupled nonlinear boundary conditions in this paper we study the global existence and the global nonexistence of doubly nonlinear degenerate parabolic systems with nonlinear boundary conditions we first prove a local existence result by the regularization method next we construct a weak comparison principle then we discuss the large time behavior of solutions by using a modified upper and lower solution methods and constructing various upper and lower solutions necessary and sufficient conditions on the global existence of all positive weak solutions are obtained

partial regularity for stationary harmonic maps at a free boundary let and be smooth riemannian manifolds of the dimension n 2 with nonempty boundary and compact without boundary we consider stationary harmonic maps u h 1 with a free boundary condition of the type u given a submanifold we prove partial boundary regularity namely sing u 0 a result that was until now only known in the interior of the domain see b the key of the proof is a new lemma that allows an extension of u by a reflection construction once the partial regularity theorem is known it is possible to reduce the dimension of the singular set further under additional assumptions on the target manifold and the submanifold

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

nonlinear fractional laplacian problems with nonlocal gradient terms let n 2 be a smooth bounded domain for s 1 2 1 we consider a problem of the form where 0 is a real parameter f belongs to a suitable lebesgue space and is a nonlocal gradient square term given by depending on the real parameter 0 we derive existence and non existence results the proof of our existence result relies on sharp calderon zygmund type regularity results for the fractional poisson equation with low integrability data we also obtain existence results for related problems involving different nonlocal diffusion terms

dynamics of the 3 d fractional complex ginzburg landau equation abstract we study the initial boundary value problem of the fractional complex ginzburg landau equation in three spatial dimensions with the dissipative effect given by a fractional laplacian a priori estimates are derived when the nonlinearity satisfies certain growth conditions using galerkin s method the existence of a global smooth solution is established uniqueness is also proved furthermore the existence of a global attractor is proved and estimates of the hausdorff and fractal dimensions for the global attractor are obtained

critical exponents in a parabolic system with inner absorption and coupled nonlinear boundary flux this paper deals with critical exponents for a parabolic system with inner absorption and coupled nonlinear boundary flux a detail analysis is given to the interactions among the multi nonlinearities in the system the critical exponent of the system is simply described via a characteristic matrix equation introduced

time space fractional stochastic ginzburg landau equation driven by fractional brownian motion abstract the current paper is devoted to the time space fractional ginzburg landau equation driven by fractional brownian motion the spatial temporal regularity of the nonlocal stochastic convolution is firstly established and then the existence and uniqueness of the global mild solution are obtained by the banach fixed point theorem and mittag leffler functions finally the numerical simulations for the time fractional stochastic ginzburg landau equation are provided to verify the mathematical analysis results

a variational model for dislocations in the line tension limit we study the interaction of a singularly perturbed multiwell energy with an anisotropic nonlocal regularizing term of h1 2 type and a pinning condition this functional arises in a phase field model for dislocations which was recently proposed by koslowski cuitino and ortiz but it is also of broader mathematical interest in the context of the dislocation model we identify the limit of the energy in all scaling regimes for the number n of obstacles the most interesting regime is n ln where is a nondimensional length scale related to the size of the crystal lattice in this case the limiting model is of line tension type one important feature of our model is that the set of energy wells is periodic and hence not compact thus a key ingredient in the proof is a compactness estimate up to a single translation for finite energy sequences which generalizes earlier results from alberti bouchitte and seppecher for the two well problem with a h1 2 regularization

an adaptive numerical method to handle blow up in a parabolic system we study numerical approximations to solutions of a system of two nonlinear diffusion equations in a bounded interval coupled at the boundary in a nonlinear way in certain cases the system develops a blow up singularity in finite time fixed mesh methods are not well suited to approximate the problem near the singularity as an alternative to reproduce the behaviour of the continuous solution we present an adaptive in space procedure the scheme recovers the conditions for blow up and non simultaneous blow up it also gives the correct non simultaneous blow up rate and set moreover the numerical simultaneous blow up rates coincide with the continuous ones in the cases when the latter are known finally we present numerical experiments that illustrate the behaviour of the adaptive method

singular reaction diffusion equations of porous medium type abstract a class of reaction diffusions equations with nonlinear boundary conditions and porous medium type of diffusion is investigated by the method of upper and lower solutions the diffusion term in the equation includes the so called slow diffusion m 1 fast diffusion 0 m 1 and super diffusion m 0 and the reaction term may involve a singular function the aim of this paper is to show the existence of a global time dependent solution the existence of positive minimal and maximal steady state solutions including the uniqueness of the positive solution and the asymptotic behavior of the time dependent solution in relation to the positive steady state solutions of particular concern is the convergence of the time dependent solution to a unique positive steady state solution and the global attraction property of the reaction diffusion problem applications are given to several specific problems which are of considerable interest in the current literature

global attractors of some predator prey reaction diffusion systems with density dependent diffusion and time delays abstract this paper deals with a two species and a three species predator prey reaction diffusion systems where the diffusion coefficients are density dependent and time delays are involved in the reaction functions the diffusion terms are of porous medium type which are degenerate and the boundary conditions are of neumann type the aim of the paper is to investigate the asymptotic behavior of the time dependent solution in relation to various steady state solutions of the system this includes the existence of a unique positive classical solution global attraction of a steady state solution and stability or instability of various semitrivial solutions and the positive steady state solution

quasilinear parabolic and elliptic systems with mixed quasimonotone functions abstract this paper deals with a class of quasilinear parabolic and elliptic systems with mixed quasimonotone reaction functions the boundary condition in the system may be dirichlet nonlinear or a combination of these two types the elliptic operators in the system are allowed to be degenerate the aim is to show the existence and uniqueness of a classical solution to the parabolic system the existence of maximal and minimal solutions or quasisolutions of the elliptic system and the asymptotic behavior of the solution of the parabolic system this consideration leads to a global attractor of the parabolic system as well as an one sided stability of the maximal and minimal solutions applications of these results are given to three models arising from biology and ecology where diffusion coefficients are density dependent and are degenerate these applications exhibit quite distinct dynamical behavior of the population species between degenerate density dependent diffusion and constant diffusion

a variational problem with lack of compactness for h1 2 s1 s1 maps of prescribed degree abstract we consider for maps in h 1 2 s 1 s 1 a family of semi norms equivalent to the standard one we ask whether for such a norm there is some map in h 1 2 s 1 s 1 of prescribed topological degree equal to 1 and minimal norm in general the answer is no due to concentration phenomena the existence of a minimal map is sensitive to small perturbations of the norm we derive a sufficient condition for the existence of minimal maps in particular we prove that for every given norm there are arbitrarily small perturbations of it for which the minimum is attained in case there is no minimizer we determine the asymptotic behavior of minimizing sequences we prove that for such minimizing sequences the energy concentrates near a point of s 1 we describe this concentration in terms of bubbling off of circles

partially constrained boundary conditions with energy minimizing mappings on considere la regularite a la frontiere pour une application minimisant l energie u m n pour deux types de conditions aux limites frontiere libre et condition limite partiellement contrainte

blow up estimates of the positive solution of a parabolic system this paper establishes the blow up estimates for the systems ut u 0 vt v 0 in br 0 t br rn with the nonlinear boundary conditions u um1vn1 and v um2vn2 on sr 0 t where 0 m1 1 m2 and 0 n2 1 n1 we prove that c t t 2 max u x t c t t 2 and c t t 2 max v x t c t t 2 under some monotonicity assumptions on the initial values where n1 n2 1 m2 m1 1 and n1m2 1 m1 1 n2

interactions among multi nonlinearities in a nonlinear diffusion system with absorptions and nonlinear boundary flux abstract this paper deals with a nonlinear diffusion system with inner absorptions and coupled nonlinear boundary conditions two parameters which solve an introduced matrix equation covering all the eight nonlinear exponents of the system are used to get a simple description for the interactions among the multi nonlinearities in the system blow up or not of the solutions is determined by the signs of the two parameters in order to obtain the critical properties of the solutions for the critical sign of the parameters some further analysis on the absorption coefficients and the geometry of is given in addition to the discussion for the eight nonlinear exponents in the system

sub criticality of non local schrodinger systems with antisymmetric potentials and applications to half harmonic maps abstract we consider non local linear schrodinger type critical systems of the type 1 1 4 v v in r where is antisymmetric potential in l 2 r so m v is an r m valued map and v denotes the matrix multiplication we show that every solution v l 2 r r m of 1 is in fact in l loc p r r m for every 2 p in other words we prove that the system 1 which is a priori only critical in l 2 happens to have a subcritical behavior for antisymmetric potentials as an application we obtain the c loc 0 regularity of weak 1 2 harmonic maps into c 2 compact sub manifolds without boundary

blow up analysis for a quasilinear parabolic system with multi coupled nonlinearities this paper deals with a quasilinear parabolic system coupled via both nonlinear reaction terms and nonlinear boundary flux as the results of the interaction among the multi coupled nonlinearities in the system some appropriate conditions for global existence and global nonexistence of solutions are determined respectively

boundary equations and regularity theory for geometric variational systems with neumann data we study boundary regularity of maps from two dimensional domains into manifolds which are critical with respect to a generic conformally invariant variational functional and which at the boundary intersect perpendicularly with a support manifold for example harmonic maps or h surfaces with a partially free boundary condition in the interior it is known by the celebrated work of riviere that these maps satisfy a system with an antisymmetric potential from which one can derive the interior regularity of the solution avoiding a reflection argument we show that these maps satisfy along the boundary a system of equations which also exhibits a nonlocal antisymmetric potential that combines information from the interior potential and the geometric neumann boundary condition we then proceed to show boundary regularity for solutions to such systems