minimization problem associated with an improved hardy sobolev type inequality abstract we consider the existence or non existence of a minimizer of the following minimization problems associated with an improved hardy sobolev type inequality introduced by ioku 2019 i a inf u w 0 1 p b r 0 b r u p d x b r u p s v a x d x p p s where v a x 1 x s 1 a x r n p p 1 1 x s where 1 p n and 0 s p the minimization problem i a is equivalent to the minimization problem associated with the classical hardy sobolev inequality on r n via a transformation if we restrict ourselves to radial functions in contrast to the classical results for a 0 we show the existence of non radial minimizers for the hardy sobolev critical exponent p s p n s n p on bounded domains furthermore as an application of the transformation we derive an infinite dimensional form of the classical sobolev inequality in some sense

gidas ni nirenberg results for finite difference equations estimates of approximate symmetry are positive solutions of finite difference boundary value problems hu f u in h u 0 on h as symmetric as the domain to answer this question we first show by examples that almost arbitrary non symmetric solutions can be constructed this is in striking difference to the continuous case where by the famous gidas ni nirenberg theorem b gidas wei ming ni l nirenberg symmetry and related problems via the maximum principle comm math phys 68 1979 209 243 positive solutions inherit the symmetry of the underlying domain then we prove approximate symmetry theorems for solutions on equidistantly meshed n dimensional cubes explicit estimates depending on the data are given which show that the solutions become more symmetric as the discretization gets finer the quality of the estimates depends on whether or not f 0 0 the one dimensional case stands out in two ways the proofs are elementary and the estimates for the defect of symmetry are o h compared to o 1 log h in the higher dimensional case

eigenfunctions of composite hermitian operators with application to discrete and continuous radiating systems some properties and applications of hermitian operators composed of any integral operator and its adjoint are studied such operators arise in array and aperture antenna theory and the eigenfunctions of the corresponding hermitian operators form complete sets for the expansion of sources and fields in their respective regions the eigenfunctions with the largest eigenvalue form a source and field pair which radiate the largest power under a fixed source norm constraint and can be used for example to maximize power transferred from an array to a point or from one aperture to a second aperture

a symmetry theorem and applications to nonlinear partial differential equations on donne des applications d un nouveau theoreme sur les proprietes de symetrie des solutions des equations differentielles non lineaires de la forme lu f x u dans u 0 ou u n 0 ou est une region bornee ou non bornee de r n

the computation of one parameter families of bifurcating elastic surfaces the problem of constructing the middle surface of a deformed elastic shell from its first and second fundamental forms hat a alpha beta and hat b alpha beta is considered the undeformed shell is a spherical cap of radius r and thickness h with an angular width 2 theta 0 where 0 theta 0 pi 2 the cap is subjected to a constant uniform load lambda and is simply supported at its edge the authors seek to compute the one parameter families of buckled states that branch from the unbuckled state of the shell this is accomplished in two steps first a finite element method is used to solve the governing shell equations a pair of fourth order nonlinear partial differential equations pdes a solution of this system is a curvature potential w a stress potential f and the load lambda using liapunov schmidt reduction it can be shown that solutions possessing a variety of symmetries bifurcate from the unbuckled state of the shell in the work presented here these

search extension method for multiple solutions of a nonlinear problem abstract based on the eigensystem j j of multiple solutions of nonlinear problem u u 3 0 min u 0 on are approximated a new method called the search extension method sem composed of three steps is proposed numerical examples in a square and the unit circle are presented to show the efficiency of our method

antenna synthesis and optimization using generalized characteristic modes a set of modes called generalized characteristic modes is described previously developed characteristic modes exhibit orthogonality properties over an obstacle surface as well as the complete sphere at infinity the generalized characteristic modes described here also exhibit orthogonality properties over the obstacle surface but rather than showing orthogonality over the sphere at infinity they are orthogonal over any region of space similar to generalized inagaki modes theories developed for antenna pattern synthesis and directivity optimization using generalized inagaki modes can be applied in an entirely parallel fashion using generalized characteristic modes comparisons are made between generalized characteristic modes and generalized inagaki modes through three examples

the heat equation with a singular potential of concern is the singular problem du dt au c x 2 iz x tz x 0 u0 x and its generalizations here c 0 x g rn t 0 and and t 0 are nonnegative and not both identically zero there is a dimension dependent constant c n such that the problem has no solution for c ct n for c s ct a necessary and sufficient conditions are found for and u0 so that a nonnegative solution exists

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

minimizers of caffarelli kohn nirenberg inequalities with the singularity on the boundary let be a bounded smooth domain in bf r n n geqq 3 and d a 1 2 omega be the completion of c 0 infty omega with respect to the norm u a 2 int omega x 2a nabla u 2 d x the caffarelli kohn nirenberg inequalities state that there is a constant c 0 such that begin array ll int omega x bq u q d x frac 2 q leqq c int omega x 2a nabla u 2 d x end array quad quad 0 1 for u in d a 1 2 omega and infty a frac n 2 2 quad 0 leqq b a leqq 1 quad q frac 2n n 2 2 b a we prove the best constant for 0 1 s a b omega inf limits u in d a 1 2 backslash 0 frac int omega x 2a nabla u 2 d x int omega x bq u q d x frac 2 q is always achieved in d a 1 2 omega provided that 0 in partial omega and the mean curvature h 0 0 where a b satisfies i a 0 rm and n geqq 4 if a 0 and 1 b 0 then the result was proved by ghoussoub and robert 12

constrained mountain pass algorithm for the numerical solution of semilinear elliptic problems a new numerical algorithm for solving semilinear elliptic problems is presented a variational formulation is used and critical points of a c1 functional subject to a constraint given by a level set of another c1 functional or an intersection of such level sets of finitely many functionals are sought first constrained local minima are looked for then constrained mountain pass points the approach is based on the deformation lemma and the mountain pass theorem in a constrained setting several examples are given showing new numerical solutions in various applications

a unified approach to improved l p hardy inequalities with best constants we present a unified approach to improved l p hardy inequalities in r n we consider hardy potentials that involve either the distance from a point or the distance from the boundary or even the intermediate case where the distance is taken from a surface of codimension 1 k n in our main result we add to the right hand side of the classical hardy inequality a weighted l p norm with optimal weight and best constant we also prove nonhomogeneous improved hardy inequalities where the right hand side involves weighted l q norms q p

an efficient and stable method for computing multiple saddle points with symmetries in this paper an efficient and stable numerical algorithm for computing multiple saddle points with symmetries is developed by modifying the local minimax method established in y li and j zhou siam j sci comput 23 2001 pp 840 865 y li and j zhou siam j sci comput 24 2002 pp 840 865 first an invariant space is defined in a more general sense and a principle of invariant criticality is proved for the generalization then the orthogonal projection to the invariant space is used to preserve the invariance and to reduce computational error across iterations simple averaging formulas are used for the orthogonal projections numerical computations of examples with various symmetries of which some can and others cannot be characterized by a compact group of linear isomorphisms are carried out to confirm the theory and to illustrate applications the mathematical features of various problems demonstrated in these examples fall into two categories nodal solutions of saddle point type with large morse indices and nonradial positive solutions via symmetry breaking in radially symmetric elliptic problems the new numerical algorithm generates these rather unstable solutions in an efficient and stable way the existence of many unstable solutions and their behavior found in this paper remain to be investigated

stability and convergence of a finite element method for a semilinear elliptical problem with small viscosity abstract in this paper we propose and analyze a new finite element method for a semilinear elliptical problem with small viscosity firstly we prove the existence and uniqueness of solution for its variational problem then we obtain the stability and convergence of the corresponding finite element galerkin approximation furthermore we derive the optimal error estimates in the l 2 and h 1 norms for the finite element approximations respectively finally several numerical experiments are provided to confirm the above theoretical results

extremal functions of generalized critical hardy inequalities abstract in this paper we show the existence and non existence of minimizers of the following minimization problems which include an open problem mentioned by horiuchi and kumlin 20 g a inf u w 0 1 n 0 u n d x u q f a x d x n q where f a x x n log a r x first we give an answer to the open problem when b r 0 next we investigate the minimization problems on general bounded domains in this case the results depend on the shape of the domain finally symmetry breaking property of the minimizers is proved for sufficiently large

high order numerical method for the nonlinear helmholtz equation with material discontinuities in one space dimension the nonlinear helmholtz equation nlh models the propagation of electromagnetic waves in kerr media and describes a range of important phenomena in nonlinear optics and in other areas in our previous work we developed a fourth order method for its numerical solution that involved an iterative solver based on freezing the nonlinearity the method enabled a direct simulation of nonlinear self focusing in the nonparaxial regime and a quantitative prediction of backscattering however our simulations showed that there is a threshold value for the magnitude of the nonlinearity above which the iterations diverge in this study we numerically solve the one dimensional nlh using a newton type nonlinear solver because the kerr nonlinearity contains absolute values of the field the nlh has to be recast as a system of two real equations in order to apply newton s method our numerical simulations show that newton s method converges rapidly and in contradistinction with the iterations based on freezing the nonlinearity enables computations for very high levels of nonlinearity in addition we introduce a novel compact finite volume fourth order discretization for the nlh with material discontinuities our computations corroborate the design fourth order convergence of the method the one dimensional results of the current paper create a foundation for the analysis of multidimensional problems in the future

analytical and numerical results for the fucik spectrum of the laplacian nontrivial solutions of u u vu 0 in rn with zero dirichlet condition on are studied the collection of the pairs v is the so called fucik spectrum f a new variational formulation for parts of f is presented and analyzed based on this formulation a minimization algorithm for the computation of a part of f is developed alternatively an approach using an implicit function argument is discussed analytically and via newton s method also numerically by combining the variational minimization method with newton s method a new bifurcation result is first observed numerically and then proved rigorously by replacing the variational minimization method by the mountain pass algorithm higher curves in f are found numerically several numerical results are discussed including a further example of the previously recorded phenomenon of crossing of fucik curves originating from different eigenvalues

antenna synthesis and optimization using weighted inagaki modes inagaki modes represent a new modal formalism for problems of radiation and scattering from arbitrary discrete and continuous structures these modes can be generalized to be orthogonal over any sector of the sphere at infinity as well as the source region itself theory and application of these modes to problems of pattern synthesis and array optimization including constraints are presented examples are given for a line source an array of printed dipoles and a dipole array in free space

existence and non existence of the first eigenvalue of the perturbed hardy sobolev operator in this paper we study the existence non existence and simplicity of the rst eigenvalue of the perturbed hardy sobolev operator i i 1 4 n i2 2 q jxj2 under various assumptions on the perturbation q we study the asymptotic behaviour of the rst eigenfunction near the origin when the perturbation q is q s 0 s 1 we will also establish the best constant in a hardy sobolev inequality proved by adimurthi e t a l

newton s method s basin of attraction for sign changing solutions of concave and convex nonlinearities abstract in this paper we will illustrate and visualize the chaotic behavior for solving the sign changing solutions of concave convex nonlinearities we propose contractor iteration algorithm and bifurcation method to improve the computation speed and avoid local convergence of newton raphson method by this method sign changing numerical solutions will be computed and visualized in various domains

bifurcation method for computing the multiple positive solutions to p henon equation in this paper we proposed the algorithms to solving the d 4 symmetric positive solutions to the boundary value problem of p henon equation taking r in p henon equation as a bifurcation parameter the symmetry breaking bifurcation point on the branch of the d 4 symmetric positive solutions is found via the extended systems based on the liapunov schmidt reduction other symmetric positive solutions are computed by the branch switching method

on antenna array design using orthogonal modes antenna array design using orthogonal modes has been extensively studied in the past the design procedure is based on the projection of an arbitrary array factor on a basis defined by a set of orthogonal modes on the array structure a design method is presented that relies on the construction of an orthogonal basis using the gram schmidt orthonormalization method moreover simplifications of the orthogonal projection are provided reducing the computational complexity of the design procedure finally the validity of the method is displayed with a design example