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

the dynamics of maps tangent to the identity and with nonvanishing index let f be a germ of a holomorphic self map of c 2 at the origin o tangent to the identity and with o as a nondicritical isolated fixed point a parabolic curve for f is a holomorphic f invariant curve with o on the boundary attracted by o under the action of f it has been shown by m abate 2001 that if the characteristic direction v p t o c 2 has residual index not belonging to q then there exist parabolic curves for f tangent to v in this paper we prove using a different method that the conclusion still holds just assuming that the residual index is not vanishing at least when f is regular along v

note on surface radiation conditions the predictions of the surface radiation condition are compared with the exact analytical answer for scattering by a soft elliptical cylinder the agreement is found to be highly satisfactory for a range of parameters which can be handled analytically numerical results are presented that substantiate the analysis the surface radiation condition is generalized to be applicable when the environment is inhomogeneous study of the particular case of a circular cylinder surrounded by an inhomogeneous medium reveals that surface boundary condition gives good results for some values of the parameters where analytic manipulation is feasible

a comparison of approximate boundary conditions and infinite element methods for exterior helmholtz problems we compare the accuracy of a variety of approximate boundary conditions abcs and two versions of an infinite element method iem for approximating solutions of exterior helmholtz problems the infinite elements are of variable order in the radial direction and differ only in whether the test functions in the weak formulation are conjugated or not we find that the accuracy of the abcs decreases with increasing frequency and that at higher frequencies their accuracy is several orders of magnitude lower than that obtained with the iems we show that the convergence rate of the conjugated element is slower than the rate for the unconjugated element and that the latter element fails to converge in the far field

a wide angle on surface radiation condition applied to scattering by spheroids a rational approximation of the pseudo differential dirichlet to neumann operator for a circular cylinder is made using zero placement for an acoustically hard circular cylinder this zero corresponds to the dominant creeping wave singularity in the residue series for the time harmonic free space surface green s function based on this operator a wide angle on surface radiation condition osrc is developed first for two dimensional slender convex bodies for the object scattering problem this osrc is accurate for wide scattering angles relative to the surface normal the notion of creeping waves is then used to develop a wide angle three dimensional osrc applicable to smooth convex bodies application to both hard and soft prolate and oblate spheroids is presented and excellent agreement is found in comparison with a more costly technique

absorbing boundary conditions for difference approximations to the multi dimensional wave equation construction des conditions aux limites absorbantes pour l equation des ondes multidimensionnelle a l aide d approximations aux differences formulation sous forme canonique des conditions aux limites absorbantes analytiques

residual indices of holomorphic maps relative to singular curves of fixed points on surfaces let m be a two dimensional complex manifold and let f m to m be a holomorphic map that fixes pointwise a possibly singular compact reduced and globally irreducible curve c subset m we give a notion of degeneracy of f at a point of c it turns out that f is non degenerate at one point if and only if it is non degenerate at every point of c when f is non degenerate on c we define a residual index for f at each point of c then we prove that the sum of the indices is equal to the self intersection number of c

on a class of semilinear fractional elliptic equations involving outside dirac data abstract the purpose of this article is to give a complete study of the weak solutions of the fractional elliptic equation 0 1 u u p 0 in b 1 e n u 0 in r n b 1 e n where p 0 with 0 1 denotes the fractional laplacian operator in the principle value sense b 1 e n is the unit ball centered at e n 0 0 1 in r n with n 2 and 0 is the dirac mass concentrated at the origin we prove that problem eq 0 1 admits a unique weak solution when p 1 2 n moreover if in addition p n 2 n 2 the weak solution vanishes as 1 we also show that problem eq 0 1 does not have any weak solution when p 0 1 2 n these results are very surprising since they are in total contradiction with the classical setting i e u u p 0 in b 1 e n u 0 in r n b 1 e n for which it has been proved that there are no solutions for p n 1 n 1

coupling of dirichlet to neumann boundary condition and finite difference methods in curvilinear coordinates for multiple scattering the applicability of the dirichlet to neumann technique coupled with finite difference methods is enhanced by extending it to multiple scattering from obstacles of arbitrary shape the original boundary value problem bvp for the multiple scattering problem is reformulated as an interface bvp a heterogenous medium with variable physical properties in the vicinity of the obstacles is considered a rigorous proof of the equivalence between these two problems for smooth interfaces in two and three dimensions for any finite number of obstacles is given the problem is written in terms of generalized curvilinear coordinates inside the computational region then novel elliptic grids conforming to complex geometrical configurations of several two dimensional obstacles are constructed and approximations of the scattered field supported by them are obtained the numerical method developed is validated by comparing the approximate and exact far field patterns for the scattering from two circular obstacles in this case for a second order finite difference scheme a second order convergence of the numerical solution to the exact solution is easily verified

nonlinear boundary value problems relative to harmonic functions abstract we study the problem of finding a function u verifying u 0 in under the boundary condition u n g u on where r n is a smooth domain n is the normal unit outward vector to is a measure on and g a continuous nondecreasing function we give sufficient condition on g for this problem to be solvable for any measure when g r r p 1 r p 1 we give conditions in order an isolated singularity on to be removable we also give capacitary conditions on a measure in order the problem with g r r p 1 r to be solvable for some we also study the isolated singularities of functions satisfying u 0 in and u n g u 0 on 0

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

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

estimates on green functions and poisson kernels for symmetric stable processes one of the most basic and most important subfamily of levy processes is symmetric stable processes a symmetric stable process x on rn is a levy process whose transition density p t x y relative to the lebesgue measure is uniquely determined by its fourier transform rn e ix p t x dx e t here must be in the interval 0 2 when 2 we get a brownian motion running with a time clock twice as fast as the standard one brownian motion plays a central role in modern probability theory and has numerous important applications in other scientific areas as well as in many other branches of mathematics thus it has been intensively studied in this paper symmetric stable processes are referred to the case when 0 2 unless otherwise specified in the last few years there has been an explosive growth in the study of physical and economic systems that can be successfully modeled with the use of stable processes stable processes are now widely used in physics operations research queuing theory mathematical finance and risk estimation in some physics literatures symmetric stable processes are called levy flights and they have been applied to a wide range of very complex physics issues such as turbulent diffusion vortex dynamics anomalous diffusion in rotating flows and molecular spectral fluctuations in mathematical finance stable processes can be used to model stock returns in incomplete market for these and more applications of stable processes please see the interesting book 14 by janicki and weron and the references therein and the recent article 15 by klafter shlesinger and zuomofen in order to make precise predictions about natural phenomena and to better cope with these

on surface radiation conditions for an ellipse we compare several on surface radiation boundary conditions in two dimensions for solving the helmholtz equation exterior to an ellipse we also introduce a new boundary condition for an ellipse based on a modal expansion in mathieu functions we compare the osrc to a finite difference method

a two dimensional polynomial mapping with a wandering fatou component we show that there exist polynomial endomorphisms of 2 possessing a wandering fatou component these mappings are polynomial skew products and can be chosen to extend holomorphically of p 2 c we also find real examples with wandering domains in 2 the proof is based on parabolic implosion techniques and is based on an original idea of m lyubich

three dimensional contour fdtd modeling of scattering from single and multiple bodies this paper introduces a generalization of the three dimensional finite difference time domain fdtd method the three dimensional contour fdtd cfdtd method is introduced the fdtd method represents curved media boundaries as stepped edges through the use of subcell modeling the cfdtd method conformably models bodies with curved surfaces yet retains the ability to model corners and edges electromagnetic scattering from single and multiple bodies is presented

an improved surface radiation condition for high frequency acoustic scattering problems a new artificial boundary condition for two dimensional acoustic scattering problems is constructed in this paper it is designed from the principal symbol of the dirichlet to neumann map so that both the propagating and evanescent fields are properly modeled in the high frequency regime the new artificial boundary condition can be used as an on surface radiation condition for efficient approximation of the wave field the accuracy and efficiency of our method are demonstrated in numerical examples

regularity theory for fully nonlinear integro differential equations we consider nonlinear integro differential equations like the ones that arise from stochastic control problems with purely jump levy processes we obtain a nonlocal version of the abp estimate harnack inequality and interior c 1 regularity for general fully nonlinear integro differential equations our estimates remain uniform as the degree of the equation approaches two so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations

improved accuracy for the helmholtz equation in unbounded domains based on properties of the helmholtz equation we derive a new equation for an auxiliary variable this reduces much of the oscillations of the solution leading to more accurate numerical approximations to the original unknown computations confirm the improved accuracy of the new models in both two and three dimensions this also improves the accuracy when one wants the solution at neighbouring wavenumbers by using an expansion in k we examine the accuracy for both waveguide and scattering problems as a function of k h and the forcing mode l the use of local absorbing boundary conditions is also examined as well as the location of the outer surface as functions of k connections with parabolic approximations are analysed copyright 2004 john wiley sons ltd

three dimensional finite element calculations in acoustic scattering using arbitrarily shaped convex artificial boundaries we report on a generalization of the bayliss gunzburger turkel non reflecting boundary conditions to arbitrarily shaped convex artificial boundaries for elongated scatterers such as submarines we show that this generalization can improve significantly the computational efficiency of finite element methods applied to the solution of three dimensional acoustic scattering problems copyright 2001 john wiley sons ltd

semilinear fractional elliptic equations with measures in unbounded domain abstract in this paper we study the existence of nonnegative weak solutions to e u h u in a general regular domain which vanish in r n where denotes the fractional laplacian with 0 1 is a nonnegative radon measure and h r r is a continuous nondecreasing function satisfying a subcritical integrability condition furthermore we analyze properties of weak solution u k to e with r n k 0 and h s s p where k 0 p 0 n n 2 and 0 denotes dirac mass at the origin finally we show for p 0 1 2 n that u k in r n as k and for p 1 2 n n n 2 that lim k u k x c x 2 p 1 with c 0 which is a classical solution of u u p 0 in r n 0

a new 3 d conformal pec fdtd scheme with user defined geometric precision and derived stability criterion a new conformal finite difference time domain cfdtd updating scheme for metallic surfaces nonaligned in the grid is presented in this paper in contrast to existing conformal models the new model can be formulated with the original yee fdtd update equation therefore the proposed scheme can be easily added in standard fdtd codes even if the codes are already parallelized or hardware accelerated in addition based on the commonly used conventional stability criterion a derivation of the stability is presented and based on the conformal geometric information a time step reduction formula is presented the time step reduction is used as a user defined parameter to tradeoff speed versus accuracy the achievable geometric precision is optimized to a given time step therefore even with the conventional time step no reduction the presented scheme profits from the conformal discretization to show the performance and the robustness of the proposed scheme canonical validations and two real world applications were investigated a broadband low profile circular antenna was successfully simulated showing the benefit of the conformal fdtd method compared to the conventional scheme furthermore a cad based mobile phone was conformally discretized and successfully simulated showing that the proposed scheme is highly suited for the simulation of advanced engineering problems

the dynamics near quasi parabolic fixed points of holomorphic diffeomorphisms in c 2 let f be a germ of holomorphic diffeomorphism of c 2 fixing o and such that df o has eigenvalues 1 and e itheta with e itheta 1 and e itheta not equal 1 introducing suitable normal forms for f we define an invariant nu f greater than or equal to 2 and a generic condition that of being dynamically separating in the case f is dynamically separating we prove that there exist nu f 1 parabolic curves for f at o tangent to the eigenspace of 1

index theorems for holomorphic self maps let m be a complex manifold and s subset m a possibly singular subvariety of m let f colon m to m be a holomorphic map such that f restricted to s is the identity we show that one can associate to f a holomorphic section x f of a sheaf related to the embedding of s in m and that such a section reads the dynamical behavior of f along s in particular we prove that under generic hypotheses the canonical section x f induces a holomorphic action in the sense of bott on the normal bundle of the regular part of s in m and this allows to obtain for holomorphic self maps with non isolated fixed points index theorems similar to camacho sad baum bott and variation index theorems for holomorphic foliations finally we apply our index theorems to obtain information about topology and dynamics of holomorphic self maps of surfaces with a compact curve of fixed points

existence of minimizers of functionals involving the fractional gradient in the absence of compactness symmetry and monotonicity abstract we establish general assumptions under which a constrained variational problem involving the fractional gradient and a local nonlinearity admits minimizers

the dirichlet elliptic problem involving regional fractional laplacian in this paper we study the solutions of elliptic equations involving regional fractional laplacian e u f in a bounded regular domain in rn n 2 with c2 boundary subject to dirichlet boundary g on where 12 1 and the operator denotes the regional fractional laplacian we prove that when g 0 problem e admits a unique weak solution under the hypotheses that f l2 f l1 dx and f m where x dist x 2 1 and m is a space of all radon measures satisfying d finally we provide an integration by parts formula for the classical solution of e with boundary data g

a poincare bendixson theorem for meromorphic connections on riemann surfaces we shall prove a poincare bendixson theorem describing the asymptotic behavior of geodesics for a meromorphic connection on a compact riemann surface we shall also briefly discuss the case of non compact riemann surfaces and study in detail the geodesics for a holomorphic connection on a complex torus

parabolic manifolds for semi attractive analytic transformations of mathbf c n we study the local dynamics of semi attractive analytic transformations of cn under certain assumptions rivi showed the existence of parabolic manifolds of dimension m 1 where m is the number of eigenvalues with modulus strictly less than one assuming moreover that certain matrix has p eigenvalues with positive real part we show the existence of parabolic manifolds of dimension m p 1

a higher order on surface radiation condition derived from an analytic representation of a dirichlet to neumann map on surface radiation conditions are useful for obtaining approximate solutions to scattering problems involving compact obstacles an analytic representation of the dirichlet to neumann map for a circle is derived and used to construct a higher order on surface radiation condition for a generally convex perfectly conducting body in two dimensions this approach is based on a hankel function in which a tangential operator appears in the index in the high frequency limit this analytic representation approaches the square root of a differential operator which commonly arises in the application of parabolic equation techniques to propagation problems treating the scattered field propagation angle relative to the surface normal and the surface curvature as independent parameters the representation is fit to a rational function to provide an accurate and efficient on surface radiation condition that is tested for various examples

singular solutions of some nonlinear elliptic equations abstract we give a classification of isolated singularities of any solution of the equation u u u q 1 0 if 0 is the singular point we prove that there exist two types of singularities when 1 n n 2 and that we have essentially the following in 0 i either u x l q n x 2 q 1 with l q n 2 q 1 2q q 1 n 1 q 1 ii or u x c x where c is any constant and the fundamental solution of the laplacian in r n

error analysis for absorbing boundary conditions an estimate is derived for the error committed by the introduction of artificial boundaries and corresponding artificial boundary conditions when solving wave equations on unbounded domains the estimate has two terms one is proportional to the largest reflection coefficient for the artificial boundary condition the maximum taken only on those rays which appear in the computation the second term is proportional to 1 k wherek is a measure of the average frequency present in the solution

comparison of local radiation boundary conditions for the scalar helmholtz equation with general boundary shapes the relative accuracy of several local radiation boundary conditions based on the second order bayliss turkel 1980 condition are evaluated these boundary conditions permit the approximate solution of the scalar helmholtz equation in an infinite domain using traditional finite element and finite difference techniques unlike the standard bayliss turkel condition the generalizations considered here are applicable to noncircular solution domains the accuracy of these conditions are investigated for elliptical and linear circular boundaries

diagonalization of nondiagonalizable discrete holomorphic dynamical systems we shall describe a canonical procedure to associate to any germ of holomorphic self map f of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i n fixing the origin so that df o is invertible and nondiagonalizable an n dimensional complex manifold m a holomorphic map m inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i n a point e m and a germ of holomorphic self map inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i of m such that restricted to m 1 o is a biholomorphism between m 1 o and inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i n o inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 05i f and e is a fixed point of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 06i such that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 07i is diagonalizable furthermore we shall use this construction to describe the local dynamics of such an f nearby the origin when sp df o 1

invariant manifolds of parabolic fixed points ii approximations by sums of homogeneous functions abstract we study the computation of local approximations of invariant manifolds of parabolic fixed points and parabolic periodic orbits of periodic vector fields if the dimension of these manifolds is two or greater in general it is not possible to obtain polynomial approximations here we develop an algorithm to obtain them as sums of homogeneous functions by solving suitable cohomological equations we deal with both the differentiable and analytic cases we also study the dependence on parameters in the companion paper 4 these approximations are used to obtain the existence of true invariant manifolds close by examples are provided

local absorbing boundary conditions for elliptical shaped boundaries we compare several local absorbing boundary conditions for solving the helmholtz equation by a finite difference or finite element method exterior to a general scatterer these boundary conditions are imposed on an artificial elliptical or prolate spheroid outer surface in order to compare the computational solution with an analytical solution we consider as an example scattering about an ellipse we solve the helmholtz equation with both finite differences and finite elements we also introduce a new boundary condition for an ellipse based on a modal expansion

an elliptical cylindrical fdtd algorithm for modeling conformal patch antenna an explicit finite difference time domain fdtd method is described using a uniform elliptical cylindrical spatial grid for time domain maxwell s equations to accurately and efficiently model conformal elliptical cylindrical line fed patch antennas and overcome stair casing errors such errors are encountered when a cartesian spatial grid is used for modeling curved surfaces in addition the method described in this paper obviates coding complexity encountered in modeling of curved surfaces which is manifest using cp fdtd or cfdtd techniques characterizing expressions representing the absorbing boundary condition and courant stability condition are presented and used to provide a highly effective algorithm to simulate conformal microstrip antennas the proposed fdtd algorithm is validated against hfss results showing excellent correlation

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

compact high order schemes with gradient direction derivatives for absorbing boundary conditions we consider several compact high order absorbing boundary conditions abcs for the helmholtz equation in three dimensions a technique called the gradient method gm for abcs is also introduced and combined with the high order abcs gm is based on the principle of using directional derivatives in the direction of the wavefront propagation the new abcs are used together with the recently introduced compact sixth order finite difference scheme for variable wave numbers experiments on problems with known analytic solutions produced very accurate results demonstrating the efficacy of the high order schemes particularly when combined with gm the new abcs are then applied to the seg eage salt model showing the advantages of the new schemes

a finite element method for solving helmholtz type equations in waveguides and other unbounded domains a finite element method is described for solving helmholtz type boundary value problems in unbounded regions including those with infinite boundaries typical examples include the propagation of acoustic or electromagnetic waves in waveguides the radiation condition at infinity is based on separation of variables and differs from the classical sommerfeld radiation condition it is shown that the problem may be replaced by a boundary value problem on a fixed bounded domain the behavior of the solution near infinity is incorporated in a nonlocal boundary condition this problem is given a weak or variational formulation and the finite element method is then applied it is proved that optimal error estimates hold

on closed invariant sets in local dynamics abstract we investigate the dynamical behaviour of a holomorphic map on an f invariant subset c of u where f u c k we study two cases when u is an open connected and polynomially convex subset of c k and c u closed in u and when u has a p s h barrier at each of its points and c is not relatively compact in u in the second part of the paper we prove a birkhoff s type theorem for holomorphic maps in several complex variables i e given an injective holomorphic map f defined in a neighborhood of u with u star shaped and f u a runge domain we prove the existence of a unique forward invariant maximal compact and connected subset of u which touches u

the boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption in the first part of the paper we establish the existence of a boundary trace for positive solutions of the equation u g x u 0 in a smooth domain n for a general class of positive nonlinearities this class includes every space independent monotone increasing g which satisfies the keller osserman condition as well as degenerate nonlinearities g q of the form g q x u d x u q 1u with 2 and q 1 the boundary trace is given by a positive regular borel measure which may blow up on compact sets in the second part we concentrate on the family of nonlinearities g q determine the critical value of the exponent q for fixed 2 and discuss a positive solutions with an isolated singularity for subcritical nonlinearities and b the boundary value problem for u g q x u 0 with boundary data given by a positive regular borel measure possibly unbounded we show that in the subcritical case the problem possesses a unique solution for every such measure 2003 wiley periodicals inc

existence and nonexistence of solutions to elliptic equations involving the hardy potential abstract the purpose of this paper is to study the nonexistence of nonnegative super solutions to the problem 0 1 u x 2 u q u p in r n k where 0 1 r p 0 k is a compact set in r n with n 1 and q is a potential in r n k satisfying that lim inf x q x x 0 for some 2 when 1 is the laplacian operator and when 0 1 it is the fractional laplacian which is a typical nonlocal operator in this paper we find the critical exponent p 1 depending on and such that problem 0 1 has no nontrivial nonnegative super solutions for 0 p p furthermore we also consider the existence and nonexistence of isolated singular solutions to the equation u x 2 u q u p in r n 0 lim x u x 0 where 0 p 0 and q x 1 x with 0 2

initial trace of positive solutions to fractional diffusion equation with absorption in this paper we prove the existence of an initial trace t u of any positive solution u of the semilinear fractional diffusion equation h t u u f t x u 0 in r times r n where n 1 where the operator with 0 1 is the fractional laplacian and f r times r n times r r is a caratheodory function satisfying f t x u u 0 for all t x u r times r n times r we define the regular set of the trace t u as an open subset of r u r n carrying a nonnegative radon measive u such that lim t 0 ru u t x x dx ru d c 2 0 r u and the singular set s u r n r u as the set points a such that lim sup t 0 b a u t x dx 0 we study the reverse problem of constructing a positive solution to h with a given initial trace s where s r n is a closed set and is a positive radon measure on r r n s and develop the case f t x u t u p where 1 and p 1

nonlocal scalar field equations qualitative properties asymptotic profiles and local uniqueness of solutions abstract we study the nonlocal scalar field equation with a vanishing parameter p s u u u p 2 u u q 2 u in r n u h s r n where s 0 1 n 2 s q p 2 are fixed parameters and 0 is a vanishing parameter for small we prove the existence and qualitative properties of positive solutions next we study the asymptotic behavior of ground state solutions when p is subcritical supercritical or critical sobolev exponent 2 2 n n 2 s for p 2 the ground state solution asymptotically coincides with unique positive ground state solution of s u u u p whereas for p 2 the asymptotic behavior of the solutions is given by the unique positive solution of the nonlocal critical emden fowler type equation for p 2 the solution asymptotically coincides with a ground state solution of s u u p u q furthermore using these asymptotic profile of positive solutions we establish the local uniqueness of positive solution

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

invariant manifolds of parabolic fixed points i existence and dependence on parameters abstract in this paper we study the existence and regularity of stable manifolds associated to fixed points of parabolic type in the differentiable and analytic cases using the parametrization method the parametrization method relies on a suitable approximate solution of a functional equation in the case of parabolic points if the manifolds have dimension two or higher in general this approximation cannot be obtained in the ring of polynomials but as a sum of homogeneous functions and it is given in 4 assuming a sufficiently good approximation is found here we provide an a posteriori result which gives a true invariant manifold close to the approximate one in the differentiable case in some cases there is a loss of regularity we also consider the case of parabolic periodic orbits of periodic vector fields and the dependence of the manifolds on parameters examples are provided we apply our method to prove that in several situations namely related to the parabolic infinity in the elliptic spatial three body problem these invariant manifolds exist and do have polynomial approximations

the fractional schrodinger equation with general nonnegative potentials the weighted space approach abstract we study the dirichlet problem for the stationary schrodinger fractional laplacian equation s u v u f posed in bounded domain r n with zero outside conditions we consider general nonnegative potentials v l l o c 1 and prove well posedness of very weak solutions when the data are chosen in an optimal class of weighted integrable functions f important properties of the solutions such as its boundary behaviour are derived the case of super singular potentials that blow up near the boundary is given special consideration since it leads to so called flat solutions we comment on related literature

holomorphic dynamics near germs of singular curves let m be a two dimensional complex manifold p m and open image in new window a germ of holomorphic foliation of m at p letopen image in new window be a germ of an irreducible possibly singular curve at p in m which is a separatrix for open image in new window we prove that if the camacho sad suwa index ind open image in new window then there exists another separatrix for open image in new window at p a similar result is proved for the existence of parabolic curves for germs of holomorphic diffeomorphisms near a curve of fixed points

a static condensation reduced basis element method complex problems abstract we extend the static condensation reduced basis element scrbe method to treat the class of parametrized complex helmholtz partial differential equations the main ingredients are i static condensation at the interdomain level ii a conforming eigenfunction port representation at the interface level iii the reduced basis rb approximation of finite element fe bubble functions at the intradomain level and iv rigorous system level error bounds which reflect rb perturbation of the fe schur complement we then incorporate these ingredients in an offline online computational strategy to achieve rapid and accurate prediction of parametric systems formed from instantiations of interoperable parametrized archetype components from a library we introduce a number of extensions with respect to the original scrbe framework first primal dual rb methods for general non symmetric complex problems second stability constant procedures for weakly coercive problems at both the interdomain level and intradomain level third treatment of non port linear functional outputs as well as functions of outputs fourth consideration of particular components and outputs relevant to acoustic applications we consider several numerical examples in acoustics in particular focused on mufflers and horns to demonstrate that the approach can handle models with many parameters and or topology variations with particular reference to waveguide like applications

wide angle one way wave equations a one way wave equation also known as a paraxial or parabolic wave equation is a differential equation that permits wave propagation in certain directions only such equations are used regularly in underwater acoustics in geophysics and as energy absorbing numerical boundary conditions the design of a one way wave equation is connected with the approximation of 1 s2 1 2 on 1 1 by a rational function which has usually been carried out by pade approximation this article presents coefficients for l2 l and other alternative classes of approximants that have better wide angle behavior for theoretical results establishing the well posedness of these wide angle equations see the work of trefethen and halpern well posedness of one way wave equations and absorbing boundary conditions math comput 47 421 435 1986

quasi parabolic analytic transformations of cn abstract let f be an analytic transformation of c n with the origin o as a quasi parabolic fixed point we will associate an invariant order f to f and study the local dynamics of f when it has a non degenerate characteristic direction v and is dynamically separating in the direction v we show that for such f there exist at least f 1 parabolic curves tangent to v at the origin we also study a non dynamically separating example in details

nonlocal elliptic equations involving measures abstract in this article we study the existence of solution for the problem u f u in u 0 in r n where 0 is a parameter 0 1 and is a radon measure a weak solution is obtained by using schauder s fixed point theorem in the case where is dirac measure the symmetry of the solution is obtained by using the moving plane method

a new method to avoid reduction of time step in the cp fdtd method a new cp wcs fdtd method is developed to avoid the time step reduction in the conformal path fdtd method the stability condition in this method is not related with the shape and size of the distorted cell and the time step size in this method is only determined by the space discretization perpendicular to the curved metal surface the cp wcs fdtd method has the same accuracy as that for the cp fdtd method and due to large time step size can be used the cp wcs fdtd has higher computation efficiency which is demonstrated by numerical examples

on accuracy conditions for the numerical computation of waves abstract the helmholtz equation k 2 n 2 u f with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems such problems can be solved numerically by first truncating the given unbounded domain imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization for example using a finite element method in practical applications the mesh size h and the wave number k are not independent but are constrained by the accuracy of the desired computation it will be shown that the number of points per wavelength measured by kh 1 is not sufficient to determine the accuracy of a given discretization for example the quantity k 3 h 2 is shown to determine the accuracy in the l 2 norm for a second order discretization method applied to several propagation models

poincare bendixson theorems for meromorphic connections and holomorphic homogeneous vector fields abstract we first study the dynamics of the geodesic flow of a meromorphic connection on a riemann surface and prove a poincare bendixson theorem describing recurrence properties and limit sets of geodesics for a meromorphic connection on p 1 c we then show how to associate to a homogeneous vector field q in c n a rank 1 singular holomorphic foliation f of p n 1 c and a partial meromorphic connection o along f so that integral curves of q are described by the geodesic flow of o along the leaves of f which are riemann surfaces the combination of these results yields powerful tools for a detailed study of the dynamics of homogeneous vector fields for instance in dimension two we obtain a description of recurrence properties of integral curves of q and of the behavior of the geodesic flow in a neighborhood of a singularity classifying the possible singularities both from a formal point of view and for generic singularities from a holomorphic point of view we also get examples of unexpected new phenomena we put in a coherent context scattered results previously known and we obtain as far as we know for the first time a complete description of the dynamics in a full neighborhood of the origin for a substantial class of holomorphic maps tangent to the identity finally as an example of application of our methods we study in detail the dynamics of quadratic homogeneous vector fields in c 2

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

nonlinear fractional elliptic systems with boundary measure data existence and a priori estimates abstract we are concerned with positive solutions of the fractional elliptic system e s u f v in s v g u in where is an arbitrary domain in r n n 2 s s 1 2 1 and f g c l o c r for some 0 1 we establish universal a priori estimate for positive solutions of e then for c 2 bounded domain we prove the existence of positive solutions of e with prescribed boundary value u and v where are positive bounded measure on and discuss regularity property of the solutions

bayliss turkel like radiation conditions on surfaces of arbitrary shape this paper addresses the extension of the bayliss turkel second order radiation condition to an arbitrarily shaped surface the derivation is based mainly on the pseudo differential calculus as well as on the introduction of a criterion providing a precise handling of the approximation process involved in the derivation of the radiation condition the radiation condition then ranges among the most accurate of those of order two as a by product of the derivation almost all known radiation conditions of order less than or equal to two are recovered and their respective accuracies are compared

a general approach for high order absorbing boundary conditions for the helmholtz equation when solving a scattering problem in an unbounded space one needs to implement the sommerfeld condition as a boundary condition at infinity to ensure no energy penetrates the system in practice solving a scattering problem involves truncating the region and implementing a boundary condition on an artificial outer boundary bayliss gunzburger and turkel bgt suggested an absorbing boundary condition abc as a sequence of operators aimed at annihilating elements from the solution s series representation their method was practical only up to a second order condition later hagstrom and hariharan hh suggested a method which used auxiliary functions and enabled implementation of higher order conditions we compare various absorbing boundary conditions abcs and introduce a new method to construct high order abcs generalizing the hh method we then derive from this general method abcs based on different series representations of the solution to the helmholtz equation in polar elliptical and spherical coordinates some of these abcs are generalizations of previously constructed abcs and some are new these new abcs produce accurate solutions to the helmholtz equation which are much less dependent on the various parameters of the problem such as the value of k or the eccentricity of the ellipse in addition to constructing new abcs our general method sheds light on the connection between various abcs computations are presented to verify the high accuracy of these new abcs

three dimensional approximate local dtn boundary conditions for prolate spheroid boundaries we propose a new class of approximate local dtn boundary conditions to be applied on prolate spheroidal shaped exterior boundaries when solving problems of acoustic scattering by elongated obstacles these conditions are a exact for the first modes b easy to implement and to parallelize c compatible with the local structure of the computational finite element scheme and d applicable to exterior ellipsoidal shaped boundaries that are more suitable in terms of cost effectiveness for surrounding elongated scatterers we investigate analytically and numerically the effect of the frequency regime and the slenderness of the boundary on the accuracy of these conditions we also compare their performance to the second order absorbing boundary condition bgt2 designed by bayliss gunzburger and turkel when expressed in prolate spheroid coordinates the analysis reveals that in the low frequency regime the new second order dtn condition dtn2 retains a good level of accuracy regardless of the slenderness of the boundary in addition the dtn2 boundary condition outperforms the bgt2 condition such superiority is clearly noticeable for large eccentricity values

a computational method for the helmholtz equation in unbounded domains based on the minimization of an integral functional abstract we study a new approach to the problem of transparent boundary conditions for the helmholtz equation in unbounded domains our approach is based on the minimization of an integral functional arising from a volume integral formulation of the radiation condition the index of refraction does not need to be constant at infinity and may have some angular dependency as well as perturbations we prove analytical results on the convergence of the approximate solution numerical examples for different shapes of the artificial boundary and for non constant indexes of refraction will be presented

fractional heat equations with subcritical absorption having a measure as initial data we study existence and uniqueness of weak solutions to f tu u h t u 0 in 0 rn with initial condition u 0 in rn where n 2 the operator is the fractional laplacian with 0 1 is a bounded radon measure and h 0 r r is a continuous function satisfying a subcritical integrability condition in particular if h t u t up with 1 and 0 p p 1 2 1 n we prove that there exists a unique weak solution uk to f with k 0 where 0 is the dirac mass at the origin we obtain that uk in 0 rn as k for p 0 1 and the limit of uk exists as k when 1 p p we denote it by u when 1 2 1 n 2 p p p u is the minimal self similar solution of f tu u t up 0 in 0 rn with the initial condition u 0 0 in rn 0 and it satisfies u 0 x 0 for x 0 while if 1 p p then u up where up is the maximal solution of the differential equation y t yp 0 on r

a simple impedance infinite element for the finite element solution of the three dimensional wave equation in unbounded domains this paper is concerned with the development of an efficient and accurate impedance infinite element that can be used either in the frequency or directly in the time domain for the modeling and solution of problems described by the scalar three dimensional wave equation in infinite or semi infinite domains the infinite domain is truncated and the effect of the truncated infinite region is simulated by the introduction of an absorbing boundary condition prescribed on the truncation boundary a systematic procedure for the construction of a family of such conditions of increasing accuracy and complexity is developed with explicit formulas given for approximations up to second order a central feature of this high order approximation is that it can be expressed within the context of a finite element formulation as a set of local infinite elements located at the boundary of the computational domain with each element defined by a pair of symmetric time invariant stiffness and damping matrices this makes it possible to incorporate readily the new local boundary element into finite element software developed for purely interior regions for applications involving steady state harmonic or transient excitations whereas the theory has been developed formally for arbitrary smooth boundary surfaces here details are provided for ellipsoidal and spherical boundaries thus far only the latter has been implemented and tested in problems involving cavities and rigid scatterers of spherical and cubic shape numerical experiments in both the frequency and time domain attest to the efficacy and accuracy of the proposed new element

real topological entropy versus metric entropy for birational measure preserving transformations abstract we consider a family of birational measure preserving transformations of two complex variables depending on one parameter for which simple rational expressions for the dynamical zeta function have been conjectured together with an equality between the topological entropy and the logarithm of the arnold complexity divided by the number of iterations similar results have been obtained for the adaptation of these two concepts to dynamical systems of real variables yielding to introduce a real topological entropy and a real arnold complexity we try to compare here the kolmogorov sinai metric entropy and this real arnold complexity or real topological entropy on this particular example of a one parameter dependent birational transformation of two variables more precisely we analyze using an infinite precision calculation the lyapunov characteristic exponents for various values of the parameter of the birational transformation in order to compare these results with the ones for the real arnold complexity we find a quite surprising result for this very birational example and in fact for a large set of birational measure preserving mappings generated by involutions the lyapunov characteristic exponents seem to be equal to zero or at least extremely small for all the orbits we have considered and for all values of the parameter birational measure preserving transformations generated by involutions could thus allow to better understand the difference between the topological description and the probabilistic description of discrete dynamical systems many birational measure preserving transformations generated by involutions seem to provide examples of discrete dynamical systems which can be topologically chaotic while they are metrically almost quasi periodic heuristically this can be understood as a consequence of the fact that their orbits seem to form some kind of transcendental foliation of the two dimensional space of variables

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