conformal deformation of a riemannian metric to a scalar flat metric with constant mean curvature on the boundary one of the most celebrated theorems in mathematics is the riemann mapping theorem it says that an open simply connected proper subset of the plane is conformally diffeomorphic to the disk in higher dimensions very few regions are conformally diffeomorphic to the ball however we can still ask whether a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways namely it has zero scalar curvature and its boundary has constant mean curvature in this paper we generalize the riemann mapping theorem to higher dimensions in that sense

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

nonresonance between the first two eigenvalues for a steklov problem abstract in this paper we study the solvability of the steklov problem p u u p 2 u in u p 2 u f x u on under assumptions on the asymptotic behaviour of the quotients f x s s p 2 s and p f x s s p which extends the classical results with dirichlet boundary conditions that for a e x the limits at the infinity of these quotients lie between the first two eigenvalues

multiple positive solutions for semilinear elliptic systems with nonlinear boundary condition abstract in this paper we study the nonlinear boundary value problem u u f u u v in v v f v u v in u n g u u v v n g v u v on where is a bounded domain in r n with smooth boundary and the pairs of f u f v and g u g v are subcritical nonlinearities with the help of the nehari manifold we show existence of at least two positive solutions when one of the pairs of f u f v and g u g v is sublinear and the other one superlinear

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

existence of solutions for dirichlet problems with p laplacian an existence theorem of the solutions is obtained for the p laplacian equation with dirichlet boundary conditions by using the critical point theorem developed in song and tang 2006 10 martinez and rossi 2003 11 where f x t 0tf x s ds is assumed to lie between the two continuous eigenvalues k k 1 k 1 of the p laplacian

a quasilinearization method for p laplacian equations with a nonlinear boundary condition abstract in this work we consider a nonlinear elliptic problem with a nonlinear boundary condition assuming the existence of an ordered couple of a subsolution and a supersolution we develop a quasilinearization method in order to construct an iterative scheme that converges to a solution

optimal liouville type theorems for noncooperative elliptic schrodinger systems and applications we study multi component elliptic schrodinger systems arising in nonlinear optics and bose einstein condensation phenomena we prove new liouville type nonexistence theorems as well as a priori bounds decay and singularity estimates this is shown under an optimal sobolev growth restriction on the nonlinearities thus improving on recent results of dancer et al and of tavares et al these systems are of non cooperative form and hence cannot be tackled by maximum principle methods such as moving planes instead we rely on a delicate combination of rellich pohozaev type identities sobolev and interpolation inequalities on sn 1 and feedback and measure arguments we also extend our results to a rather general class of gradient type systems

multiple positive solutions of a nonlinear boundary value problem involving a sign changing weight abstract in this paper we study the multiplicity of positive solutions for the following elliptic equation u u 0 in r n u a x u q 2 u b x u p 2 u on r n where 1 q 2 p 2 2 2 n 1 n 2 if n 3 2 if n 2 r n x x n r n 1 r x n 0 is an upper half space in r n 0 and the functions a and b are satisfying some suitable conditions using the decomposition of the nehari manifold we prove that the equation has at least two positive solutions provided is sufficiently small

the complete extension of the biot tolstoy solution to the density contrast wedge with sample calculations the biot tolstoy exact time domain solution for the three dimensional impulse response of an impenetrable wedge is extended to accommodate the isovelocity or density contrast wedge fourier transformation of the time and axial variables along with a kantorovich lebedev transform applied to the cylindrical radial coordinate leads to a solution in terms of residue series when the wedge angle is a rational fraction of the residue series can be reduced to a finite sum which is evaluated for some special cases the total pressure field consists of geometrical acoustics contributions as predicted by snell s laws plus a modified version of the biot tolstoy diffraction field

the best constant for the sobolev trace embedding from w1 1 ω into l1 ω abstract in this paper we study the best constant 1 for the trace map from w 1 1 into l 1 we show that this constant is attained in bv when 1 1 moreover we prove that this constant can be obtained as limit when p 1 of the best constant of w 1 p l p to perform the proofs we will look at neumann problems involving the 1 laplacian 1 u div du du

a convex concave problem with a nonlinear boundary condition in this paper we study the existence of nontrivial solutions of the problem u u u p 2uin u u q 2uon with 1 q 2 n 1 n 2 and 1 p 2n n 2 in the concave convex case i e 1 q 2 p if is small there exist two positive solutions while for large there is no positive solution when p is critical and q subcritical we obtain existence results using the concentration compactness method finally we apply the implicit function theorem to obtain solutions for small near u0 1

existence and non existence results for a quasilinear problem with nonlinear boundary condition abstract we study the problem div a x u p 2 u 1 x 1 u q 2 u h x u r 2 u in r n a x u p 2 u n b x u p 2 u g x u on u 0 in where is an unbounded domain with smooth boundary n denotes the unit outward normal vector on and 0 are real parameters we assume throughout that p q r p pn n p 1 p n n 1 q n p p n while a b and h are positive functions we show that there exist an open interval i and 0 such that the problem has no solution if i and 0 furthermore there exist an open interval j i and 0 0 such that for any j the above problem has at least a solution if 0 but it has no solution provided that 0 0 our paper extends previous results obtained by j chabrowski and k pfluger

acoustic diffraction by deformed edges of finite length theory and experiment the acoustic diffraction by deformed edges of finite length is described analytically and in the frequency domain through use of an approximate line integral formulation originally applied to deformed finite length cylinders t k stanton j acoust soc am 86 691 705 1989 in this case of edges the factor to the integral specific to the scattering per unit length by the cylinder is replaced by one specific to edge diffraction the formulation is applied to laboratory data involving elastic disks of finite thickness by incorporating an approximate double edge diffraction formula for three sided semi infinite barriers based on modified solutions of pierce a d pierce j acoust soc am 55 941 955 1974 the diffraction in the backscatter direction by individual elastic disks was measured using a variety of disks spanning a range of diameters thicknesses and two material properties a broadband acoustic source was used and the disks rotated over 360 in 1 increments given the overlap betw

existence of infinitely many weak solutions for the p laplacian with nonlinear boundary conditions abstract in this paper we deal with the existence of weak solutions for quasilinear elliptic problem involving a p laplacian of the form p u x u p 2 u f x u in u p 2 u u p 2 u on we consider the above problem under several conditions on f for f superlinear and subcritical with respect to u we prove the existence of infinitely many solutions of the above problem by using the fountain theorem and the dual fountain theorem respectively for the case where f is critical with a subcritical perturbation namely f x u u p 2 u u r 2 u we show that there exists at least a nontrivial solution when p r p and there exist infinitely many solutions when 1 r p by using the mountain pass theorem and the concentration compactness principle respectively

eigenvalues of the p x laplacian steklov problem abstract consider steklov eigenvalue problem involving the p x laplacian on a bounded domain the open subset of r n with n 2 as follows p x u u p x 2 u in u p x 2 u u p x 2 u on where p x constant we prove that the existence of infinitely many eigenvalue sequences unlike the p laplacian case for a variable exponent p x constant there does not exist a principal eigenvalue and the set of all eigenvalues is not closed under some assumptions finally we present some sufficient conditions for the infimum of all eigenvalues is zero and positive respectively

self similar solutions of the porous medium equation in a half space with a nonlinear boundary condition existence and symmetry abstract we find existence of a nonnegative compactly supported solution of the problem u u in r n u u on r n moreover we prove that every nonnegative solution with finite energy is compactly supported and radially symmetric in the tangential variables

on the fucik spectrum and a resonance problem for the p laplacian with a nonlinear boundary condition in this paper we prove that there exists a first curve of the fucik spectrum of the problem pu u p 2u in with a nonlinear boundary condition given by u p 2 u u p 1 u p 1 on the boundary of the domain we also prove that there exists a sequence of curves of the fucik spectrum which exist locally in the neighborhood of suitable eigenvalues of the p laplacian with a nonlinear boundary condition finally we study a resonance problem with respect to the fucik spectrum

impulse response of density contrast wedge using normal coordinates the exact impulse response of a point source to a rigid wedge was first derived by biot and tolstoy in normal coordinates i tolstoy and c s clay ocean acoustics theory and experiment in underwater sound american institute of physics new york 1987 appendix 5 the solution is compact and includes both reflected and diffracted waves explicitly kinney et al j acoust soc am 73 183 194 1983 introduced the solution of free wedge pressure release which is similar to the rigid case except sign changes for certain terms the solutions have been verified in laboratory experiments however neither model can be applied directly to many actual cases such as ocean floor where the ratio of density of the base sediment to that of the water is neither infinite ridge nor zero pressure release the normal coordinate solution presented here is the impulse response to an isovelocity c c and density contrast wedge it is an extension to the solution given by biot and tolstoy the eigenva

nonexistence existence and multiplicity of positive solutions to the p x laplacian nonlinear neumann boundary value problem abstract we study the nonexistence existence and multiplicity of positive solutions for the nonlinear neumann boundary value problem involving the p x laplacian of the form p x u u p x 2 u f x u in u p x 2 u g x u on where is a bounded smooth domain in r n p c 1 and p x 1 for x using the sub supersolution method and the variational principles under appropriate assumptions on f and g we prove that there exists 0 such that the problem has at least two positive solutions if has at least one positive solution if and has no positive solution if

resonance problems for the p laplacian systems abstract two existence theorems of the solutions are obtained for the p laplacian systems at resonance under a landesman lazer type condition by critical point theory

uniqueness of ground states of some coupled nonlinear schrodinger systems and their application abstract we establish the uniqueness of ground states of some coupled nonlinear schrodinger systems in the whole space we firstly use schwartz symmetrization to obtain the existence of ground states for a more general case to prove the uniqueness of ground states we use the radial symmetry of the ground states to transform the systems into an ordinary differential system and then we use the integral forms of the system more interestingly as an application of our uniqueness results we derive a sharp vector valued gagliardo nirenberg inequality

radial symmetry results for fractional laplacian systems abstract in this paper we generalize the direct method of moving planes for the fractional laplacian to the system case considering a coupled nonlinear system with fractional laplacian we first establish a decay at infinity principle and a narrow region principle using these principles we then obtain two radial symmetry results for the decaying solutions of the fractional laplacian systems finally we apply our method to fractional schrodinger systems and fractional henon systems

some existence theorems for elliptic resonant problems some existence theorems are obtained for the elliptic resonant problems at either the first eigenvalue or the higher eigenvalues with unbounded nonlinearity by using the minimax methods in critical point theory

note on a paper by g rowlands chaotic trajectories of ordinary differential equations this note discusses the possible use of perturbation methods in studying chaotic trajectories of ordinary differential equations with particular focus on a recent paper on this topic by rowlands ibid vol 16 p 585 1983

bound state solutions for the supercritical fractional schrodinger equation abstract we prove the existence of positive solutions for the supercritical nonlinear fractional schrodinger equation s u v x u u p 0 in r n with u x 0 as x where p n 2 s n 2 s for s 0 1 n 2 s we show that if v x o x 2 s as x then for p n 2 s 1 n 2 s 1 this problem admits a continuum of solutions more generally for p n 2 s n 2 s conditions for solvability are also provided this result is the extension of the work by davila del pino musso and wei to the fractional case our main contributions are the existence of a smooth radially symmetric entire solution of s w w p in r n and the analysis of its properties the difficulty here is the lack of phase plane analysis for a nonlocal ode instead we use conformal geometry methods together with schaaf s argument as in the paper by ao chan delatorre fontelos gonzalez and wei on the singular fractional yamabe problem

resonance problems for p q laplacian systems abstract some existence results of weak solutions are established for p q laplacian systems which are resonant at an arbitrary eigenvalue by the minimax methods in critical point theory

three positive solutions of semilinear elliptic equations in the half space with a hole abstract in this paper assume that h is nonnegative and h l 2 0 we prove that if h l 2 is sufficiently small then there are at least three positive solutions of eq 1 in r n d where d is a c 1 1 bounded domain in r n

an optimization problem for nonlinear steklov eigenvalues with a boundary potential abstract in this paper we analyze an optimization problem for the first nonlinear steklov eigenvalue plus a boundary potential with respect to the potential function which is assumed to be uniformly bounded and with fixed l 1 norm

resonance problems for the p laplacian with a nonlinear boundary condition abstract consider resonance problems at an arbitrary eigenvalue of the elliptic equations p u u p 2 u g 1 u h 1 x in u p 2 u u p 2 u g 2 u h 2 x on some existence theorems are obtained under the landesman lazer type conditions by using variational method

on a p kirchhoff equation via fountain theorem and dual fountain theorem abstract in this paper we show the existence of infinite solutions to the kirchhoff type quasilinear elliptic equation m u p x u p d x p 1 p u x u p 2 u f x u in a smooth bounded domain r n with nonlinear boundary condition u p 2 u u p 2 on the method we used here is based on fountain theorem and dual fountain theorem

positive solutions of elliptic equations with nonlinear boundary conditions abstract this paper deals with the nonlinear elliptic equation u u f x u in a bounded smooth domain r n with a nonlinear boundary value condition the existence results are obtained by the sub supersolution method and the mountain pass lemma and nonexistence is also considered

solutions for neumann boundary value problems involving p x laplace operators abstract in this paper we study the nonlinear neumann boundary value problem of the following form p div u p x 2 u u p x 2 u f x u in u p x 2 u g x u on using the variational method under appropriate assumptions on f and g we obtain a number of results on existence and multiplicity of solutions also what is also interesting is that we obtain some results which can be considered as extensions of the classical result named combined effects of concave and convex nonlinearities moreover the positive solution of the problem is considered

the liouville equation in a half plane abstract we classify the solutions of the equation u a e u 0 in the half plane r 2 that satisfy the neumann boundary condition u t c e u 2 on r 2 an analogous problem in the once punctured disc d r 2 is also solved

remarks on multiple nontrivial solutions for quasi linear resonant problems in this paper morse theory and local linking are used to study the existence of multiple nontrivial solutions for a class of dirichlet boundary value problems with double resonance at infinity and at 0

the impulse response of an aperture numerical calculations within the framework of the wedge assemblage method a numerical calculation technique applicable to arbitrary aperture shapes on which dirichlet soft or neumann hard boundary conditions exist is described the method is based on a generalization of the wedge assemblage wa method to fully two dimensional surfaces and yields the impulse response directly benchmark tests of the technique are conducted for the case of a circular aperture the results are in near perfect agreement with the reference solution provided by a t matrix code in addition to a solid validation of the extended wa method it is shown in this paper that multiple scattering effects can be accurately included finally the physical insight that the wa method offers is demonstrated through its ability to completely and accurately dissect the scattered field into reflected diffracted and multiple diffracted components

multiple solution results for elliptic neumann problems involving set valued nonlinearities abstract the main goal of this paper is to present multiple solution results for elliptic inclusions of clarke s gradient type under nonlinear neumann boundary conditions involving the p laplacian and set valued nonlinearities to be more precise we study the inclusion p u f x u u p 2 u in with the boundary condition u p 2 u a u p 1 b u p 1 g x u on we prove the existence of two constant sign solutions and one sign changing solution depending on the parameters a and b our approach is based on truncation techniques and comparison principles for elliptic inclusions along with variational tools like the nonsmooth mountain pass theorem the second deformation lemma for locally lipschitz functionals as well as comparison results of local c 1 minimizers and local w 1 p minimizers of nonsmooth functionals

a quasilinearization method for elliptic problems with a nonlinear boundary condition abstract we study a nonlinear elliptic second order problem with a nonlinear boundary condition assuming the existence of an ordered couple of a supersolution and a subsolution we develop a quasilinearization method in order to construct an iterative scheme that converges to a solution furthermore under an extra assumption we prove that the convergence is quadratic

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

addendum conformal deformation of a riemannian metric to a scalar flat metric with constant mean curvature is finite on the other hand the argument at the beginning of proposition 2 1 in e shows that if al b is finite then q m om is finite jin zhiren pointed out to me that the sobolev quotient q m o9m can be oo this is the case if we delete a small geodesic ball on a compact manifold without boundary with negative scalar curvature more generally the sobolev quotient is oo if the first eigenvalue for the conformal laplacian with respect to dirichlet boundary condition is negative in order to see that let p1 be the first eigenfunction for the problem

positive solutions of some semilinear elliptic equations in r n with neumann boundary conditions abstract in this paper we consider positive classical solutions of u f u x x t r n n 3 u t h u on r n where f s is locally bounded in 0 and f s s 1 4 is non increasing in 0 for some n 2 h s is locally bounded in 0 and h s s 1 2 is non decreasing in 0 for the same we get that the possible solution only depends on t and several corollaries that include previous results of various authors are established

solvability for two point boundary value problems abstract suppose that h l 1 0 g c r r and lim t g t t 0 with the saddle point theorem the solvability is proved for the two point boundary value problemproblem formula under the condition that formula where formula and formula

optimal boundary holes for the sobolev trace constant in this paper we study the problem of minimizing the sobolev trace rayleigh quotient u w1 p p u lq p among functions that vanish in a set contained on the boundary of given boundary measure we prove existence of extremals for this problem and analyze some particular cases where information about the location of the optimal boundary set can be given moreover we further study the shape derivative of the sobolev trace constant under regular perturbations of the boundary set

symmetry properties for positive solutions to some elliptic equations in sector domains with large amplitude abstract in this paper we study symmetry properties for positive solutions of semilinear elliptic equation u f u 0 with mixed boundary condition in a spherical sector r where the amplitude of the sector is between and 2 under certain conditions on f u we prove that all positive solutions are radially symmetric about the origin unlike well known results of b gidas et al 1979 comm math phys 68 209 243 on dirichlet problem or early results of berestycki and pacella on the same problem for r with an acute angle extra conditions on f u are needed as pointed out early by h berestycki and f pacella 1989 j funct anal 87 177 211

higher order acoustic diffraction by edges of finite thickness generalized temporal and spectral solutions of acoustic diffraction by edges of finite thickness involving higher orders diffraction are presented the solution in the time domain is an extension of medwins double edge second order diffraction solution h medwin et al j acoust soc am 72 1005 1013 1982 which is based on the huygens principle while the solution in the frequency domain is an extension of the asymptotic formulas for the double edge second order diffraction via amplitude and phase matching given by pierce a d pierce j acoust soc am 55 943 955 1974 since both solutions are approximate they have different advantages and disadvantages the contribution from higher order diffraction depends strongly on the thickness and geometry of the edges comparison between the theory and experimental diffraction data demonstrates that for edges of finite thickness the first order diffraction alone under estimates the total diffraction field significantly especially when the flat en

symmetry results for elliptic schrodinger systems on half spaces abstract in this paper we prove some symmetry and monotonicity results for classical positive solutions for elliptic schrodinger systems on half spaces with the neumann type condition our methods employ the alexandrov serrin method of moving planes combined with hopf s lemma at a corner

on an open problem involving the p x laplacian a further study on the multiplicity of weak solutions to p x laplacian equations abstract by using a variational method and the truncation technique we show the multiplicity of weak solutions to a class of p x laplacian equations an open problem proposed by professor xianling fan is solved and more general results are given in this paper

resonance problems for the p laplacian abstract we consider resonance problems at an arbitrary eigenvalue of the p laplacian and prove the existence of weak solutions assuming a standard landesman lazer condition we use variational arguments to characterize certain eigenvalues and then to establish the solvability of the given boundary value problem

necessary and sufficient conditions of existence for a system involving the p laplacian 1 p n in this work we study the existence of positive solutions for the system p pu f x u v inrn pv g x u v inrn lim x u x 0 lim x v x where pu div u p 2 u is the p laplacian 1 in some case we can have because these parameters depend on p n and the growth of f and g the existence result is related to a sub super solution methods and the nonexistence result is a consequence of picone s identity

variational approach to semilinear elliptic equation with nonlinear boundary abstract we consider the existence of positive solutions for a certain class of semilinear elliptic equation with nonlinear boundary our approach is based on the variational methods

an approximate analytic solution of the lorenz equations multiple time perturbation theory is applied to the lorenz equations to obtain approximate solutions these solutions are used to obtain the form of the mapping relating two successive maxima of one of the variables of the equations good agreement is obtained with the numerical results presented by lorenz

solvability of the forced duffing equation at resonance some existence theorems are obtained for periodic solutions of the forced duffing equation at resonance by the minimax methods

multiple positive solutions for a quasilinear elliptic equation with critical exponential nonlinearity abstract let be a bounded domain in r n n 2 with c 2 boundary in this work we study the existence of multiple positive solutions of the following problem p div u 2 n 2 1 u 2 n u u 2 n 2 1 u 2 n u u p e u n n 1 u 0 in u 2 n 2 1 u 2 n u u q on where 0 p 2 n 1 q 0 n 1 and is a non negative holder continuous function we show that there exists 0 such that p admits at least two solutions in w 1 n for 0 and no solution for

existence results for the p laplacian with nonlinear boundary conditions in this paper we study the existence of nontrivial solutions for the problem pu u p 2u in a bounded smooth domain rn with a nonlinear boundary condition given by u p 2 u f u on the boundary of the domain the proofs are based on variational and topological arguments

problemes de neumann non lineaires sur les varietes riemanniennes nonlinear neumann problems on riemannian manifolds let m g be a c compact riemannian manifold of dimension n 2 whose boundary b is an n 1 dimensional submanifold and let m m b be the interior of m study of neumann problems of the form x 0 in m d dn g y 0 on b where for every t x y r m b t x and g t y are bounded by c 1 t a or c exp t a application to the determination of a conformal metric for which the scalar curvature of m and the mean curvature of b take prescribed values

liouville type theorems for nonlinear elliptic equations on extended grushin manifolds abstract in this paper we study two liouville theorems for nonlinear boundary value problems on extended grushin manifolds these problems are closely related to nonlinear fractional grushin equations we will prove that these problems do not possess nontrivial positive solution under some assumptions on the nonlinear terms we use the moving plane method in an integral form to prove our results

positive solutions of nonlinear problems involving the square root of the laplacian abstract we consider nonlinear elliptic problems involving a nonlocal operator the square root of the laplacian in a bounded domain with zero dirichlet boundary conditions for positive solutions to problems with power nonlinearities we establish existence and regularity results as well as a priori estimates of gidas spruck type in addition among other results we prove a symmetry theorem of gidas ni nirenberg type

symmetry results for semilinear elliptic systems in the whole space in this paper we obtain symmetry results of gidas ni nirenberg type for cooperative nonlinear elliptic systems in the whole space

a geometric inequality and a symmetry result for elliptic systems involving the fractional laplacian we study the symmetry properties for solutions of elliptic systems of the type s1u f1 u v s2v f2 u v where f cloc1 1 r2 s1 s2 0 1 and the operator s is the so called fractional laplacian we obtain some poincare type formulas for the harmonic extension in the half space that we use to prove a symmetry result both for stable and for monotone solutions

second order asymptotical regularization methods for inverse problems in partial differential equations abstract we develop second order asymptotical regularization soar methods for solving inverse source problems in elliptic partial differential equations with both dirichlet and neumann boundary data we show the convergence results of soar with the fixed damping parameter as well as with a dynamic damping parameter which is a continuous analog of nesterov s acceleration method moreover by using morozov s discrepancy principle together with a newly developed total energy discrepancy principle we prove that the approximate solution of soar weakly converges to an exact source function as the measurement noise goes to zero a damped symplectic scheme combined with the finite element method is developed for the numerical implementation of soar which yields a novel iterative regularization scheme for solving inverse source problems several numerical examples are given to show the accuracy and the acceleration effect of soar a comparison with the state of the art methods is also provided

the nehari manifold for a semilinear elliptic problem with the nonlinear boundary condition abstract using the nehari manifold and fibering maps we prove the existence theorem of the nonlinear boundary problem u u u p 2 u for x u n u q 2 u for x on a bounded domain r n we discuss how the nehari manifold changes as changes and show how the existence of solutions depends on the properties of the nehari manifold

a local mountain pass theorem and applications to a double perturbed p x laplacian equations in this paper we prove a local mountain pass theorem without p s condition using this theorem and ricceri s variational principle we consider a double perturbed neumann problem with nonlinear boundary condition of the email protected p x u a x u p x 2u f x u email protected 1 x u email protected u p x email protected email protected email protected g x u email protected 2 x u email protected email protected least seven solutions are obtained under different assumptions

existence and multiplicity results for quasilinear elliptic exterior problems with nonlinear boundary conditions abstract in this paper we study a class of quasilinear elliptic exterior problems with nonlinear boundary conditions existence of ground states and multiplicity results are obtained via variational methods

strong resonance for some quasilinear elliptic equations abstract we study the existence of the weak solutions of the nonlinear boundary value problem p u 1 u p 2 u g u h x in u 0 on where the nonlinearity g is of the landesman lazer type our sufficient conditions generalize all previously published results about the strong resonance at the first eigenvalue

noise attenuation by a hard wedge shaped barrier this paper is concerned with the problem of sound screening by a wedge like barrier the sound source is assumed to be point like and the receiver is located in the shadow of the source sound field so that according to geometrical optics only the field diffracted by the edge of the barrier is considered first for the hard wedge in space three models are used for calculating the amplitude of the edge diffracted field these are the uniform theory of diffraction utd the hadden pierce model both in the frequency domain and the biot tolstoy theory of diffraction which is a time domain formulation it is first shown that even at relatively low frequencies the frequency domain models perform quite satisfactorily as compared to the exact time domain theory hence and due to its relative simplicity the utd is proposed as an accurate calculation scheme for solving problems with edge diffraction by hard wedges it is also proved from theoretical calculations that the amplitude of the edge diffracted field increases for an increasing angle of the wedge and consequently the hard half plane gives the lowest field amplitude in the shadow zone some applications are then considered for evaluating the performance of a barrier on a flat ground either completely hard or with mixed homogeneous boundary conditions an improvement of the scheme for calculating the sound field in the all hard case is achieved through considering the multiple diffraction in this case only to the second order between the top of the wedge barrier and its base the results show that for usually occurring situations increasing the angle of the hard wedge barrier affects negatively its efficiency through diminishing its insertion loss these conclusions are also supported by the results of some experimental measurements conducted at a scale model level

an analytic secondary source model of edge diffraction impulse responses a new impulse response model for the edge diffraction from finite rigid or soft wedges is presented which is based on the exact biot tolstoy solution the new model is an extension of the work by medwin et al h medwin et al j acoust soc am 72 1005 1013 1982 in that the concept of secondary edge sources is used it is shown that analytical directivity functions for such edge sources can be derived and that they give the correct solution for the infinite wedge these functions support the assumption for the first order diffraction model suggested by medwin et al that the contributions to the impulse response from the two sides around the apex point are exactly identical the analytical functions also indicate that medwin s second order diffraction model contains approximations which however might be of minor importance for most geometries access to analytical directivity functions makes it possible to derive explicit expressions for the first and even second order diffraction for certain geometries an example of this is axisymmetric scattering from a thin circular rigid or soft disc for which the new model gives first order diffraction results within 0 20 db of published reference frequency domain results and the second order diffraction results also agree well with the reference results scattering from a rectangular plate is studied as well and comparisons with published numerical results show that the new model gives accurate results it is shown that the directivity functions can lead to efficient and accurate numerical implementations for first and second order diffraction

existence and nonexistence of solutions for quasilinear elliptic equations abstract in this paper we study the problem p u f x u u in u 0 on where r n is a smooth bounded domain n 2 and p u div u p 2 u defines the p laplacian we provide some necessary and sufficient conditions on f under which the problem admits a weak solution for the case p 2 we obtain more general conditions on f the main ingredients are degree theory and the super subsolution method

exact explicit solutions for diffraction by hard sound barriers and seamounts a new method of obtaining exact explicit solutions for the diffraction of sound waves by truncated wedges is given which in conjunction with a previous study i tolstoy ieee j ocean eng in press 1989 can be applied to problems involving two parallel vertices and arbitrary wedge angles w w at these vertices the method combines exact explicit representations of the field diffracted by a single vertex with the classic multiple scatter algorithm to formulate the interaction between the line vertices the previous study had demonstrated the method only for cases in which w w were irrational fractions of as well as for the rather special case w w 2 i e for the hard strip in the latter case the calculations were carried out for wavenumbers k such that 1 kl 16 where l was the strip width and were shown to agree with experimental measurements h medwin et al j acoust soc am 72 1005 1013 1982 the present article extends the theory to cover the important remaining cases

a multiplicity result for generalized laplacian systems with multiparameters abstract in this paper we consider the existence nonexistence and multiplicity of a positive solution for a gelfand type generalized laplacian system with a singular indefinite weight and a vector parameter by using the upper and lower solution method and fixed point index theory we obtain a global multiplicity result with respect to the parameter

the behavior of the best sobolev trace constant and extremals in thin domains in this paper we study the asymptotic behavior of the best sobolev trace constant and extremals for the immersion w 1 p o l q o in a bounded smooth domain when it is contracted in one direction we find that the limit problem when rescaled in a suitable way is a sobolev type immersion in weighted spaces over a projection of o w 1 p p o a l q p o b for the special case p q this problem leads to an eigenvalue problem with a nonlinear boundary condition we also study the convergence of the eigenvalues and eigenvectors in this case