caffarelli kohn nirenberg type inequalities of fractional order with applications let 0 1p 1 be such that ps nps n assume that is a bounded domain containing the origin starting from the ground state inequality by r frank and r seiringer in 14 to obtain 1 the following improved hardy inequality for p 2p 2 for all q pq p there exists a positive constant c c q n s c c q n s such that rn rn u x u y p x y n psdxdy n p s rn u x p x pdx c u x u y p x y n qsdxdy for all u c0 rn here n p s n p s is the optimal constant in the hardy inequality 1 2 2 define ps pnn ps and let n ps2 then rn rn u x u y p x y n ps x y dydx s n p s rn u x ps x 2 ps pdx pps for all u c0 where s s n p s 0s s n p s 0 3 if n ps2 as a consequence of the improved hardy inequality we obtain that for all q pq p there exists a positive constant c c such that rn rn u x u y p x y n ps x y dydx c u x ps q x 2 ps q pdx pps q for all u c0 where ps q pnn qs notice that the previous inequalities can be understood as the fractional extension of the callarelli kohn nirenberg inequalities in 8

global solutions to a model of structural phase transitions in shape memory alloys abstract global in time existence and uniqueness of solutions to a nonlinear system of coupled evolution equations is discussed the system represents a model of dynamical structural phase transitions martensitic transformations in shape memory alloys

global stability of wavefronts with minimal speeds for nonlocal dispersal equations with degenerate nonlinearity abstract this paper is concerned with the global stability of wavefronts with minimal speeds for degenerate nonlocal dispersal equations by the method of super and subsolutions and a squeezing technique the wavefront with minimal speed is proved to be globally asymptotically stable with phase shift

semilinear fractional elliptic equations involving measures abstract we study the existence of weak solutions to e u g u in a bounded regular domain in r n n 2 which vanish in r n where denotes the fractional laplacian with 0 1 is a radon measure and g is a nondecreasing function satisfying some extra hypotheses when g satisfies a subcritical integrability condition we prove the existence and uniqueness of weak solution for problem e for any measure in the case where is a dirac measure we characterize the asymptotic behavior of the solution when g r r k 1 r with k supercritical we show that a condition of absolute continuity of the measure with respect to some bessel capacity is a necessary and sufficient condition in order e to be solved

global attractor for a nonlinear thermoviscoelastic model with a non convex free energy density abstract this paper is concerned with the existence of a global attractor for a semiflow governed by the weak solutions to a nonlinear one dimensional thermoviscoelasticity with a non convex free energy density the constitutive assumptions for the helmholtz free energy include the model for the study of martensitic phase transitions in shape memory alloys to describe physically phase transitions between different configurations of crystal lattices we work in a framework in which the strain u belongs to l new approaches are introduced and more delicate estimates are derived to establish the crucial l estimate of strain u in deriving the compactness of the orbit of the semiflow and the existence of an absorbing set

how to approximate the heat equation with neumann boundary conditions by nonlocal diffusion problems we present a model for nonlocal diffusion with neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary we study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero we prove that the solutions of this family of problems converge to a solution of the heat equation with neumann boundary conditions

continuum of solutions for an elliptic problem with critical growth in the gradient abstract we consider the boundary value problem p u h 0 1 l u c x u x u 2 h x where r n n 3 is a bounded domain with smooth boundary it is assumed that c 0 c h belong to l p for some p n 2 and that l we explicitly describe a condition which guarantees the existence of a unique solution of p when 0 and we show that these solutions belong to a continuum the behaviour of the continuum depends in an essential way on the existence of a solution of p 0 it crosses the axis 0 if p 0 has a solution otherwise it bifurcates from infinity at the left of the axis 0 assuming that p 0 has a solution and strengthening our assumptions to x 1 0 and h 0 we show that the continuum bifurcates from infinity on the right of the axis 0 and this implies in particular the existence of two solutions for any 0 sufficiently small

on the principal eigenvalue of some nonlocal diffusion problems abstract in this paper we analyze some properties of the principal eigenvalue 1 of the nonlocal dirichlet problem j u x u x u x in with u x 0 in r n here is a smooth bounded domain of r n and the kernel j is assumed to be a c 1 compactly supported even nonnegative function with unit integral among other properties we show that 1 is continuous or even differentiable with respect to continuous differentiable perturbations of the domain we also provide an explicit formula for the derivative finally we analyze the asymptotic behavior of the decreasing function 1 when the dilatation parameter 0 tends to zero or to infinity

semilinear fractional elliptic equations with gradient nonlinearity involving measures we study the existence of solutions to the fractional elliptic equation e1 delta alpha u epsilon g nabla u nu in a bounded regular domain omega of r n n ge2 subject to the condition e2 u 0 in omega c where epsilon 1 or 1 delta alpha denotes the fractional laplacian with alpha in 1 2 1 nu is a radon measure and g r mapsto r is a continuous function we prove the existence of weak solutions for problem e1 e2 when g is subcritical furthermore the asymptotic behavior and uniqueness of solutions are described when nu is dirac mass g s s p p geq 1 and epsilon 1

asymptotic behavior of solutions of reaction diffusion equations with nonlocal boundary conditions abstract this paper is concerned with some dynamical property of a reaction diffusion equation with nonlocal boundary condition under some conditions on the kernel in the boundary condition and suitable conditions on the reaction function the asymptotic behavior of the time dependent solution is characterized in relation to a finite or an infinite set of constant steady state solutions this characterization is determined solely by the initial function and it leads to the stability and instability of the various steady state solutions in the case of finite constant steady state solutions the time dependent solution blows up in finite time when the initial function in greater than the largest constant solution also discussed is the decay property of the solution when the kernel function in the boundary condition prossesses alternating sign in its domain

the peierls nabarro and benjamin ono equations abstract an intimate connection between the peierls nabarro equation in crystal dislocation theory and the travelling wave form of the benjamin ono equation in hydro dynamics is uncovered it is used to prove the essential uniqueness of peierls solution of the peierls nabarro equation and to give in closed form all solutions of the analogous periodic problem the latter problem is shown to be an example of global bifurcation with no secondary symmetry breaking bifurcations for a nonlinear neumann boundary value problem or equivalently for an equation involving the conjugate operator which is the hilbert transform of functions on the unit circle

nonlocal equations with measure data we develop an existence regularity and potential theory for nonlinear integrodifferential equations involving measure data the nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional p laplacean operator with measurable coefficients we introduce a natural function class where we solve the dirichlet problem and prove basic and optimal nonlinear wolff potential estimates for solutions these are the exact analogs of the results valid in the case of local quasilinear degenerate equations established by boccardo and gallouet j funct anal 87 149 169 1989 partial differ equ 17 641 655 1992 and kilpelainen and mal ann scuola norm sup pisa cl sci iv 19 591 613 1992 acta math 172 137 161 1994 as a consequence we establish a number of results that can be considered as basic building blocks for a nonlocal nonlinear potential theory fine properties of solutions calderon zygmund estimates continuity and boundedness criteria are established via wolff potentials a main tool is the introduction of a global excess functional that allows us to prove a nonlocal analog of the classical theory due to campanato ann mat pura appl iv 69 321 381 1965 our results cover the case of linear nonlocal equations with measurable coefficients and the one of the fractional laplacean and are new already in such cases

regularity of solutions to the parabolic fractional obstacle problem in recent years there has been an increasing interest in studying constrained variational problems with a fractional diffusion one of the motivations comes from mathematical finance jumpdiffusion processes where incorporated by merton 14 into the theory of option evaluation to introduce discontinuous paths in the dynamics of the stock s prices in contrast with the classical lognormal diffusion model of black and scholes 2 these models allow to take into account large price changes and they have become increasingly popular for modeling market fluctuations both for risk management and option pricing purposes let us recall that an american option gives its holder the right to buy a stock at a given price prior but not later than a given time t 0 if v x represents the rational price of an american option with a payoff at time t 0 then v will solve in the viscosity sense the following obstacle problem min lv v 0 v t

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

a nonlocal dispersal equation arising from a selection migration model in genetics this paper is concerned with the existence uniqueness and asymptotic stability of positive steady states for a nonlocal dispersal equation arising from selection migration models in genetics due to the lack of compactness and regularity of the nonlocal operators many classical methods cannot be used directly to the nonlocal dispersal problems this motivates us to find new techniques we first establish a criterion on the stability and instability of steady states this result is effective to get a necessary condition to guarantee a positive steady state it also gives the uniqueness then we prove the existence of nontrivial solutions by the corresponding auxiliary equations and maximum principle finally we consider the dynamic behavior of the initial value problem

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

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

global regularity for the free boundary in the obstacle problem for the fractional laplacian we study the regularity of the free boundary in the obstacle problem for the fractional laplacian under the assumption that the obstacle varphi satisfies delta varphi leq 0 near the contact region our main result establishes that the free boundary consists of a set of regular points which is known to be a n 1 dimensional c 1 alpha manifold by the results obtained by caffarelli salsa and silvestre it invent math 2008 and a set of singular points which we prove to be contained in a union of k dimensional c 1 submanifold k 0 ldots n 1 such a complete result on the structure of the free boundary proved by l a caffarelli it acta math 1977 and it j fourier anal appl 1998 was known only in the case of the classical laplacian and it is new even for the signorini problem which corresponds to the particular case of the 1 over 2 fractional laplacian a key ingredient behind our results is the validity of a new non degeneracy condition sup b r x 0 u varphi geq c r 2 valid at all free boundary points x 0

the dirichlet boundary problem for a nonlocal cahn hilliard equation abstract we study the existence uniqueness and continuous dependence on initial data of the solution for a nonlocal cahn hilliard equation with dirichlet boundary condition on a bounded domain under a nondegeneracy assumption the solutions are classical but when this is relaxed the equation is satisfied in a weak sense also we prove that there exists a global attractor in some metric space

the neumann boundary problem for a nonlocal cahn hilliard equation we study the existence uniqueness and continuous dependence on initial data of the solution to a nonlocal cahn hilliard equation on a bounded domain the equation generates a gradient flow for a free energy functional with nonlocal interaction also we apply a nonlinear poincare inequality to show the existence of an absorbing set in each constant mass affine space

solvability of uniformly elliptic fully nonlinear pde we get existence uniqueness and non uniqueness of viscosity solutions of uniformly elliptic fully nonlinear equations of the hamilton jacobi bellman isaacs type with unbounded ingredients and quadratic growth in the gradient without hypotheses of convexity or properness some of our results are new even for equations in divergence form

dynamics of a nonlocal dispersal sis epidemic model with neumann boundary conditions abstract in this paper we study a nonlocal dispersal susceptible infected susceptible sis epidemic model with neumann boundary condition where the spatial movement of individuals is described by a nonlocal convolution diffusion operator the transmission rate and recovery rate are spatially heterogeneous and the total population number is constant we first define the basic reproduction number r 0 and discuss the existence uniqueness and stability of steady states of the nonlocal dispersal sis epidemic model in terms of r 0 then we consider the impacts of the large diffusion rates of the susceptible and infectious populations on the persistence and extinction of the disease the obtained results indicate that the nonlocal movement of the susceptible or infectious individuals will enhance the persistence of the infectious disease in particular our analytical results suggest that the spatial heterogeneity tends to boost the spread of the infectious disease

non linear ground state representations and sharp hardy inequalities we determine the sharp constant in the hardy inequality for fractional sobolev spaces to do so we develop a non linear and non local version of the ground state representation which even yields a remainder term from the sharp hardy inequality we deduce the sharp constant in a sobolev embedding which is optimal in the lorentz scale in the appendix we characterize the cases of equality in the rearrangement inequality in fractional sobolev spaces

traveling waves in a nonlocal dispersal population model with age structure abstract this paper is concerned with the traveling waves in a single species population model which is derived by considering the nonlocal dispersal and age structure if the birth function is monotone then the existence of traveling wavefront is reduced to the existence of a pair of super and subsolutions without the requirement of smoothness it is proved that the traveling wavefront is strictly increasing and unique up to a translation the asymptotic behavior of traveling wavefronts is also obtained if the birth function is not monotone the existence of traveling wave solution is affirmed by introducing two auxiliary nonlocal dispersal equations with quasi monotonicity

radial fractional laplace operators and hessian inequalities abstract in this paper we deduce a formula for the fractional laplace operator s on radially symmetric functions useful for some applications we give a criterion of subharmonicity associated with s and apply it to a problem related to the hessian inequality of sobolev type r n k k 1 u k 1 d x c r n u f k u d x where f k is the k hessian operator on r n 1 k n 2 under some restrictions on a k convex function u in particular we show that the class of u for which the above inequality was established in ferrari et al 5 contains the extremal functions for the hessian sobolev inequality of x j wang 1994 15 this is proved using logarithmic convexity of the gaussian ratio of hypergeometric functions which might be of independent interest

some remarks on elliptic problems with critical growth in the gradient abstract in this work we analyze existence nonexistence multiplicity and regularity of solution to problem 1 u u u 2 f x in u 0 on where is a continuous nondecreasing positive function and f belongs to some suitable lebesgue spaces

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

global existence and asymptotic behavior in nonlinear thermoviscoelasticity abstract we study global existence uniqueness and asymptotic behavior as time tends to infinity of weak solutions to the system of nonlinear thermoviscoelasticity various boundary conditions are considered it is shown that for any initial data u 0 v 0 0 l w 1 h 1 there is a unique global solution u v deformation gradient velocity temperature such that u c 0 l v c 0 w 1 l 0 w 1 c 0 h 1 the constitutive assumptions for the helmholtz free energy include the models for the study of phase transition problems in shape memory alloys

identifiability of the landau ginzburg potential in a mathematical model of shape memory alloys abstract the nonlinear partial differential equations considered here arise from the conservation laws of linear momentum and energy and describe structural phase transitions in one dimensional shape memory solids with non convex landau ginzburg free energy potentials in this article the theories of analytic semigroups and real interpolation spaces for maximal accretive operators are used to show that the solutions of the model depend continuously on the admissible parameters also we show that the non physical parameters that define the free energy are identifiable from the model

asymptotic behavior for nonlocal dispersal equations abstract this paper is concerned with the existence and asymptotic behavior of solutions of a nonlocal dispersal equation by means of super subsolution method and monotone iteration we first study the existence and asymptotic behavior of solutions for a general nonlocal dispersal equation then we apply these results to our equation and show that the nonnegative solution is unique and the behavior of this solution depends on parameter in equation for 1 the solution decays to zero as t while for 1 the solution converges to the unique positive stationary solution as t in addition we show that the solution blows up under some conditions

boundary fluxes for nonlocal diffusion abstract we study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary this problem may be seen as a generalization of the usual neumann problem for the heat equation first we prove existence uniqueness and a comparison principle next we study the behavior of solutions for some prescribed boundary data including blowing up ones finally we look at a nonlinear flux boundary condition

fractional kpz equations with critical growth in the gradient respect to hardy potential abstract in this work we study the existence of positive solution to the fractional quasilinear problem s u u x 2 s u p f in u 0 in u 0 in r n where is a c 1 1 bounded domain in r n n 2 s 0 1 2 s 1 and 0 n s is defined in 3 we assume that f is a non negative function with additional hypotheses as we will see there are deep differences with respect to the case 0 more precisely if 0 there exists a critical exponent p s such that for p p s there is no positive solution moreover p s is optimal in the sense that if p p s there exists a positive solution for suitable data and sufficiently small

equivalence of weak and viscosity solutions in fractional non homogeneous problems we establish the equivalence between the notions of weak and viscosity solutions for non homogeneous equations whose main operator is the fractional p laplacian and the lower order term depends on x u and d s p u being the last one a type of fractional derivative