preconditioning and boundary conditions without h 2 estimates l 2 condition numbers and the distribution of the singular values this work deals with the behavior in the l sub 2 norm of the condition number and distribution of the l sub 2 singular values of the preconditioned operators b sub h sup minus 1 a sub h and a sub h b sub h sup minus 1 sub h where a sub h and b sub h are finite element discretizations of second order elliptic operators a and b in an earlier work manteuffel and parter proved that b sub h sup minus 1 a sub h a sub h b sub h sup minus 1 have a uniformly bounded l sub 2 condition number if and only if a sup and b sup a and b have the same boundary conditions this earlier work used the h sub 2 regularity of a and b as well as optimal l sub 2 error estimates and a quasi uniform grid for the finite element spaces in the present paper the authors first extend these condition number results to the case in which neither h sub 2 regularity and hence optimal l sub 2 error estimates nor the quasi uniformity assumption need to be satisfied instead it is assumed that the principle part of the preconditioning operator b is a scalar multiple 1 mu of the principle part of a

finite difference preconditioning cubic spline collocation method of elliptic equations we discuss a finite difference preconditioner for the c 1 interpolatory cubic spline collocation method for a uniformly elliptic operator a defined by au delta u a 0 u in omega the unit square with homogeneous dirichlet boundary conditions using the generalized field of values arguments we discuss the eigenvalues of the preconditioned matrix hat l n 2 1 hat a n 2 where hat a n 2 is the matrix of the collocation discretization operator a n 2 corresponding to a and hat l n 2 is the matrix of the finite difference operator l n 2 corresponding to the uniformly elliptic operator l given by l v delta v v in omega with homogeneous dirichlet boundary conditions finally we mention a bound of h 1 singular values of hat l 1 n 2 hat a n 2 for a general elliptic operator au delta u a 1 u x a 2 u y a 0 u in omega

an abelian category of motivic sheaves abstract a category of motivic sheaves is constructed over a variety in characteristic 0 using nori s method although the relationship with many alternative constructions remains to be clarified it does have many of the properties one expects for example it is abelian and has betti hodge and l adic realizations and it has a tannakian subcategory of motivic local systems

algebraic cycles and higher k theory here g x is the grothendieck group of coherent sheaves on x 13 gr refers to the graded group defined by the y filtration on g x cf kratzer 14 soult 20 and ch x is the chow group of codimension i algebraic cycles defined by fulton 9 the left hand isomorphism is a formal consequence of the existence of a i structure on g x while the existence of r is the central theme of the b f m rr theorem the main purpose of this paper is to define a theory of higher chow groups ch x n n 2 0 so as to obtain isomorphisms

the grothendieck duality theorem via bousfield s techniques and brown representability grothendieck proved that if f x y is a proper morphism of nice schemes then rf has a right adjoint which is given as tensor product with the relative canonical bundle the original proof was by patching local data deligne proved the existence of the adjoint by a global argument and verdier showed that this global adjoint may be computed locally in this article we show that the existence of the adjoint is an immediate consequence of brown s representability theorem lit follows almost as imme diately by smashing arguments that the adjoint is given by tensor product with a dualising complex verdier s base change theorem is an easy conse quence department of mathematics university of virginia charlottesville virginia 22903 e mail address an3rqvirginia edu this content downloaded from 157 55 39 27 on wed 07 sep 2016 06 26 19 utc all use subject to http about jstor org terms

global solution branches for equations involving nonhomogeneous operators of p laplace type abstract it is shown that if is not an eigenvalue of an associated p laplacian then the equation div x u u p 2 u f x u u with nonhomogeneous which is assumed to behave asymptotically as the function generating the associated p laplacian has a global branch of solutions u also the case of modified p laplace operators and generalizations thereof are discussed

first eigenvalue of the p laplacian under integral curvature condition abstract we give various estimates of the first eigenvalue of the p laplace operator on closed riemannian manifold with integral curvature conditions

preconditioning cubic spline collocation discretizations of elliptic equations this work considers the uniformly elliptic operator a defined by au delta u a 1 u x a 2 u y a 0 u in omega the unit square with boundary conditions u 0 on gamma 0 and partial u over partial nu alpha u on gamma 1 and its discretization based on hermite cubic spline spaces and collocation at the gauss points using an interpolatory basis with support on the gauss points one obtains the matrix hat a n h 1 over n we discuss the h 1 condition numbers and the distribution of tilde beta n singular values of the preconditioned matrices tilde beta n 1 hat a n where tilde beta n is the stiffness matrix associated with the finite element discretization of the positive definite uniformly elliptic operator b given by b v delta v b 0 v in omega with boundary conditions v 0 on gamma 0 partial v over partial nu beta v on gamma 1 the finite element space is either the space of continuous functions which are bilinear on the rectangles determined by gauss points or the space of continuous functions which are linear on the triangles of the triangulation of omega using the gauss points when a b we obtain results on the eigenvalues of tilde beta n 1 hat a n in the general case we obtain bounds and clustering results on the tilde beta n singular values of tilde beta n 1 hat a n these results are related to the results of manteuffel and parter mp parter and wong pw and wong w for finite element discretizations as well as the results of parter and rothman pr for discretizations based on legendre spectral collocation

triangulated categories of relative 1 motives abstract the category da 1 s of relative cohomological 1 motives is the localizing subcategory of the triangulated category da s da e t s q of relative voevodsky motives with rational coefficients over a scheme s which is generated by cohomological motives of curves over s we construct and study a candidate for the standard motivic t structure on da 1 s for s noetherian finite dimensional and excellent we show this t structure is non degenerate and relate its heart mm 1 s with deligne 1 motives over s in particular when s is regular the category of deligne 1 motives embeds in mm 1 s fully faithfully we also study the inclusion of da 1 s into the larger category da coh s of relative cohomological motives on s and prove that its right adjoint 1 preserves compact objects

polynomial interpolation results in sobolev spaces abstract in this paper we derive new results related to the lagrange polynomial interpolation on the nodes of gauss type quadrature formulas in particular we obtain asymptotic error estimates of the same order as the distance to the best fit in the polynomial space

the n motivic t structures for n 0 1 and 2 abstract for a field k and an integer n 0 1 2 we construct a t structure t 0 m n k t 0 m n k on voevodsky s triangulated category of motives dm eff k which we call the n motivic t structure when n 0 this is simply the usual homotopy t structure but for n 1 2 these are new t structures we will show that the category of deligne s 1 motives can be embedded as a full subcategory in the heart of the 1 motivic t structure by a rather straightforward analogy we are led to specify a class of objects in the heart of the 2 motivic t structure which we call mixed 2 motives we will also check that these objects form an abelian category

existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations in connection with the maximizing problem for the functional r u u lq u lp in w01 p 0 we consider the equation div u p 2 u x u q 2 u x x 1 p resp q p e has no nontrivial resp nonnegative nontrivial solution belonging to the class p u w01 p lq xi u q 2u lp p 1 i 1 2 n w01 p provided that is star shaped the crucial point of the proof of our result is to obtain an l estimate of weak solutions and to verify a certain pohozaev type inequality for weak solutions belonging to p

motivic periods and grothendieck arithmetic invariants abstract we construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers for the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective showing that a suitable betti de rham realization of 1 motives is fully faithful we can verify this period conjecture in several cases the divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical grothendieck period conjecture for algebraic cycles on smooth and proper schemes these divisibility properties are treated in an appendix by b kahn extending previous work of bloch and colliot thelene raskind

a global bifurcation for nonlinear equations with nonhomogeneous part abstract we study the following boundary value problem with a nonhomogeneous principal part div u u 0 u p 2 u q x u u in with the dirichlet boundary condition under certain assumptions on and q when 0 is not an eigenvalue of the p laplacian the aim of this paper is to show a bifurcation result on a noncompact component of solutions for the above nonlinear equation

global nontrivial bifurcation of homogeneous operators with an application to the p laplacian we prove the existence of a global solution branch of nontrivial solutions for a class of equations by a blow up method in particular positively homogeneous problems and equations with the p laplace operator are considered

mixed hodge complexes on algebraic varieties using the theory of mixed hodge modules we introduce the notion of mixed hodge complex on an algebraic variety and establish the relation between the filtered complex of du bois and the corresponding complex of mixed hodge modules some application to the du bois singularity is given

bifurcation problems for the laplacian in ⁿ in this paper we consider the bifurcation problem div ivulp 2vu ag x julp 2u f a x u in rn with p 1 we show that a continuum of positive solutions bifurcates out from the principal eigenvalue a1 of the problem div ivuip 2vu ag x ulp 2u here both functions f and g may change sign

the filtered ogus realisation of motives abstract we construct the filtered ogus realisation of voevodsky motives over a number field k this realisation extends the functor defined on 1 motives by andreatta barbieri viale and bertapelle as an illustration we note that the analogue of the tate conjecture holds for k3 surfaces

orthogonal spline collocation methods for partial differential equations abstract this paper provides an overview of the formulation analysis and implementation of orthogonal spline collocation osc also known as spline collocation at gauss points for the numerical solution of partial differential equations in two space variables advances in the osc theory for elliptic boundary value problems are discussed and direct and iterative methods for the solution of the osc equations examined the use of osc methods in the solution of initial boundary value problems for parabolic hyperbolic and schrodinger type systems is described with emphasis on alternating direction implicit methods the osc solution of parabolic and hyperbolic partial integro differential equations is also mentioned finally recent applications of a second spline collocation method modified spline collocation are outlined

positive solutions for a class of elliptic equations abstract in this paper the existence and bifurcation of positive solutions for a class of elliptic equations involving the p laplacian are studied by using the sub and super solution method and the topological degree argument

deligne s conjecture on 1 motives we reformulate a conjecture of deligne on 1 motives by using the integral weight filtration of gillet and soule on cohomology and prove it this implies the original conjecture up to isogeny if the degree of cohomology is at most two we can prove the conjecture for the hodge realization without isogeny and even for 1 motives with torsion

the neron severi group of a proper seminormal complex variety we prove a lefschetz 1 1 theorem for proper seminormal varieties over the complex numbers

global bifurcation for nonlinear equations abstract using a bifurcation result on noncompact branches of solutions in an abstract setting we establish the existence of global bifurcation for the following nonlinear equation div a u p 2 u 0 b u p 2 u q x u u subject to dirichlet boundary conditions under certain assumptions on a b and q when 0 is not an eigenvalue of the above divergence form

exact multiplicity for some quasilinear elliptic dirichlet problems where the non linearity changes sign abstract the exact multiplicity of positive solutions is obtained for a class of quasilinear elliptic dirichlet problems where the non linearity changes sign meanwhile the uniqueness of positive ground state on r n is also obtained

fast direct solvers for piecewise hermite bicubic orthogonal spline collocation equations fast direct methods are proposed for the solution of linear systems arising when orthogonal spline collocation with piecewise hermite bicubics is employed for the approximate solution of poisson s

global bifurcation for equations involving nonhomogeneous operators in an unbounded domain abstract we study the structure of the set of solutions of a nonlinear equation involving nonhomogeneous operators div x u u 0 g x u p 2 u f x u in r n satisfying certain conditions on g and f when 0 is not an eigenvalue of the p laplacian in some sense this is based on a bifurcation result on noncompact connected sets of solutions for nonlinear operator equations

on the shafarevich and tate conjectures for hyperkahler varieties 1 problems and results 2 polarized hyperk ihler varieties and cubic fourfoids 3 the period mapping 4 the kuga satake construction 5 the kuga satake construction in the relative setting 6 proof of theorem 1 5 1 7 proof of theorem 1 6 1 8 descent 9 proof of theorem 1 3 1 appendix 1 spinorial shimura varieties appendix 2 a short proof of the hodge conjecture for cubic fourfolds 205 210 212 216 220 224 229 233 240 244 245

preconditioning legendre spectral collocation approximations to elliptic problems this work deals with the h 1 condition numbers and the distribution of the tilde beta n m singular values of the preconditioned operators tilde beta n m 1 w n m hat a n m hat a n m is the matrix representation of the legendre spectral collocation discretization of the elliptic operator a defined by au delta u a 1 u x a 2 u y a 0 u in omega the unit square with boundary conditions u 0 on gamma 0 frac partial u partial nu a alpha u on gamma 1 tilde beta n m is the stiffness matrix associated with the finite element discretization of the positive definite elliptic operator b defined by bv delta v b 0 v in omega with boundary conditions v 0 on gamma 0 frac partial v partial nu b beta v on gamma 1 the finite element space is either the space of continuous functions which are bilinear on the rectangles determined by the legendre gauss lobatto lgl points or the s

global bifurcation of the p laplacian and related operators abstract we prove some global bifurcation theorems of rabinowitz type for a large class of operators the results apply in particular to equations involving the p laplacian

variation of mixed hodge structure i around 1970 griffiths introduced the notion of a variation of hodge structure on a complex manifold s see 17 w 2 it constitutes the axiomatization of the features possessed by the local systems of cohomology associated to a family of compact k ihler manifolds esp smooth projective varieties parametrized by s these are for a variation of hodge structure u of weight m 0 a locally constant sheaf u of q vector spaces or u of real vector spaces 1 hodge filtration horizontality a decreasing filtration o r of gs by holomorphic subbundles such that the evident connection v on takes o p into q 2 hodge decompositions if gs denotes the sheaf of functions on s and if

a local bifurcation theorem for degenerate elliptic equations with radial symmetry abstract in this work we provide local bifurcation results for equations involving the p laplacian in balls we analyze the continua c n of radial solutions emanating from n p 0 n p being the radial eigenvalues of p first we show that the only nontrivial solutions close to n p 0 lie on a continuous curve thus extending the crandall rabinowitz theorem second it is proved that c n n p 0 splits into two unbounded connected pieces characterized by their nodal properties thus sharpening previous results

analysis on eigenvalues for preconditioning cubic spline collocation method of elliptic equations abstract in the work of solving a uniformly elliptic differential equations au u a 1 u x a 2 u y a 0 u f in the unit square with boundary conditions by the c 1 cubic spline collocation method one may need to investigate efficient preconditioning techniques for this purpose using the generalized field of values argument we show the uniform bounds of the eigenvalues of the preconditioned matrix when a full finite element preconditioning is considered

a homotopic deformation along p of a leray schauder degree result and existence for u p 2u ƒ t u 0 u 0 u t 0 p 1 soit f 0 1 r r une fonction continue et soit o p r r definie par o p s s p 2 s pour p fixe 1 on considere le probleme de dirichlet suivant o p u f t u 0 u 0 u t 0 ou t est un nombre reel positif

bifurcation and admissible solutions for the hessian equation abstract we study the following eigenvalue problem of k hessian equation s k d 2 u k f u in b u 0 on b global bifurcation result is established for this problem as applications of the bifurcation result we determine the intervals of for the existence nonexistence uniqueness and multiplicity of radially symmetric k admissible solutions for this problem

p laplacian problems with jumping nonlinearities we consider the p laplacian boundary value problem 1 p u x f x u x u x a e x 0 1 2 c00u 0 c01u 0 c10u 1 c11u 1 where p 1 is a fixed number p s s p 2s s r and for each j 0 1 cj0 cj1 0 the function f 0 1 r2 r is a caratheodory function satisfying for x s t 0 1 r2 x p s e x s t f x s t x p s e x s t s 0 where l1 0 1 and e has the form e x s t x e s t with l1 0 1 0 e 0 and limr e r r1 p 0 this condition allows the nonlinearity in 1 to behave differently as u such a nonlinearity is often termed jumping related to 1 2 is the problem 3 p u a p u b p u p u in 0 1 together with 2 where a b l1 0 1 r and u x max u x 0 for x 0 1 this problem is positively homogeneous and jumping values of for which 2 3 has a nontrivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having certain nodal properties and we obtain certain spectral and degree theoretic properties of the set of half eigenvalues these properties lead to existence and nonexistence results for the problem 1 2 we also consider a related bifurcation problem and obtain a global bifurcation result similar to the well known rabinowitz global bifurcation theorem this then leads to a multiplicity result for 1 2 when the functions a and b are constant the set of half eigenvalues is closely related to the fucik spectrum of the problem and equivalent solvability results are obtained using the two approaches however when a and b are not constant the half eigenvalue approach yields stronger results

global bifurcation of the p laplacian in rn abstract we study global bifurcation for the following nonlinear equation div a x u p 2 u 0 g x u p 2 u f x u in r n satisfying certain conditions on a g and f when 0 is not an eigenvalue of the above divergence form it is based on a bifurcation result on noncompact components of solutions for nonlinear operator equations

preconditioning nonconforming finite element methods for treating dirichlet boundary conditions ii this work deals with theh 1 condition numbers and the distribution of theb h singular values of the preconditioned operators b h 1 a h 0 h 1 wherea h andb h are finite element discretizations of second order elliptic operators a andb respectively b is also assumed to be self adjoint and positive definite for conforming finite elements parter and wong have shown that the singular values cluster in a positive finite interval goldstein also has derived results on the spectral distribution ofb h 1 a h using a different approach as a generalization of the results of parter and wong the current work includes nonconforming finite element methods which deal with dirichlet boundary conditions it will be shown that in this more general setting the singular values also cluster in a positive finite interval in particular if the leading part ofb is the same as the leading part ofa then the singular values cluster about the point 1 two specific methods are given as applications of this theory they are the penalty method of babuska and the method of nearly zero boundary conditions of nitsche finally it will be shown that the same results can be proven by an approach generalized from the work of goldstein

on weight truncations in the motivic setting abstract using punctual gluing of t structures we construct an analogue of s morel s weight truncation functors for certain weight profiles in the setting of motivic sheaves as an application we construct a canonical motivic analogue of the intersection complex for an arbitrary threefold over any field k we are also able to recover certain invariants of singularities motivically

unilateral global bifurcation phenomena and nodal solutions for p laplacian in this paper we establish a dancer type unilateral global bifurcation result for one dimensional p laplacian problem p u m t p u g t u t 0 1 u 0 u 1 0 under some natural hypotheses on the perturbation function g 0 1 r r r we show that k p is a bifurcation point of the above problem and there are two distinct unbounded continua ck and ck consisting of the bifurcation branch ck from k p 0 where k p is the k th eigenvalue of the linear problem corresponding to the above problem as the applications of the above result we study the existence of nodal solutions for the following problem p u f t u 0 t 0 1 u 0 u 1 0 moreover based on the bifurcation result of girg and takac 2008 13 we prove that there exist at least a positive solution and a negative one for the following problem div p u f x u in u 0 on

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

exponential decay of c 1 cubic splines vanishing at two symmetric points in each knot interval in the course of working on the preconditioning of cubic collocation at gauss points one has to deal with the exponential decay of certain cubic splines such results were obtained by kim and parter in the case of uniform spacing in this paper we extend these results to arbitrary grids and state its application to the preconditioned cubic collocation method

bifurcation phenomena associated to the p laplace operator on determine la structure de l ensemble des solutions u de u x p 2 u x x f u u p 2 u sur 0 1 telles que u 0 u 1 0 ou p 1 et r on demontre que les solutions avec k zeros sont uniques quand 1 2

nori 1 motives let mathsf ehm be nori s category of effective homological mixed motives in this paper we consider the thick abelian subcategory mathsf ehm 1 subset mathsf ehm generated by the i th relative homology of pairs of varieties for i in 0 1 we show that mathsf ehm 1 is naturally equivalent to the abelian category t mathcal m 1 of 1 motives with torsion this is our main theorem along the way we obtain several interesting results firstly we realize t mathcal m 1 as the universal abelian category obtained using nori s formalism from the betti representation of an explicit diagram of curves secondly we obtain a conceptual proof of a theorem of vologodsky on realizations of 1 motives thirdly we verify a conjecture of deligne on extensions of 1 motives in the category of mixed realizations for those extensions that are effective in nori s sense

generalized 1 motivic sheaves abstract let k be a field of characteristic 0 we extend the definition of the category of 1 motivic sheaves introduced by barbieri viale and kahn allowing quotients of connected algebraic k groups by formal k groups we show that its bounded derived category is equivalent to the bounded derived category of the category of generalized 1 motives with torsion

eigenvalue analysis for the p laplacian convective perturbation abstract in this paper a detailed analysis of the eigenvalue problem under convection u p 2 u c u p 2 u u p 2 u 0 u 0 u 1 0 d d x is performed the analysis is based on a complete study of the phase space of the family of equations u p 2 u a u p 2 u b u p 2 u 0 a b constants

ogus realization of 1 motives abstract after introducing the ogus realization of 1 motives we prove that it is a fully faithful functor more precisely following a framework introduced by ogus considering an enriched structure on the de rham realization of 1 motives over a number field we show that it yields a full functor by making use of an algebraicity theorem of bost

a preconditioned conjugate gradient method for nonselfadjoint or indefinite orthogonal spline collocation problems we study the computation of the orthogonal spline collocation solution of a linear dirichlet boundary value problem with a nonselfadjoint or an indefinite operator of the form lu sum a ij x u x ix j sum b i x u x i c x u we apply a preconditioned conjugate gradient method to the normal system of collocation equations with a preconditioner associated with a separable operator and prove that the resulting algorithm has a convergence rate independent of the partition step size we solve a problem with the preconditioner using an efficient direct matrix decomposition algorithm on a uniform n n partition the cost of the algorithm for computing the collocation solution within a tolerance epsilon is o n 2 ln n ln epsilon

preconditioning c1 lagrange polynomial spline collocation method of elliptic equations by finite element method this paper studies a preconditioning polynomial spline collocation method for an elliptic differential equation by a finite element stiffness matrix for which we first show that the decay rate of the c 1 lagrange polynomial spline increases with increasing degree and second show that the distribution of eigenvalues for the established preconditioning system is well bounded in terms of the number of unknowns for a fixed degree of polynomial spline

crystalline realizations of 1 motives we consider the crystalline realization of deligne s 1 motives in positive characteristics and prove a comparison theorem with the de rham realization of formal liftings to zero characteristic we then show that one dimensional crystalline cohomology of an algebraic variety defined by forcing universal cohomological descent via de jong s alterations coincides with the crystalline realization of the picard 1 motive over perfect fields of characteristic 2

global bifurcation and convex solutions for the monge ampere equation abstract we study the following monge ampere equation det d 2 u 1 n f u in u 0 on by bifurcation technique we establish some results about the existence nonexistence uniqueness and multiplicity of convex solutions for this problem our results generalize and improve many important known results from previous literature

sharp de rham realization we introduce the sharp universal extension of a 1 motive with additive factors and torsion over a field of characteristic zero we define the sharp de rham realization t by passing to the lie algebra over the complex numbers we then show a sharp de rham comparison theorem in the category of formal hodge structures for a free 1 motive along with its cartier dual we get a canonical connection on their sharp extensions yielding a perfect pairing on sharp realizations thus we show how to provide one dimensional sharp de rham cohomology h dr1 of algebraic varieties

preconditioning chebyshev spectral collocation method for elliptic partial differential equations in this paper we analyze a preconditioning technique for the solution of chebyshev spectral collocation equations with dirichlet boundary conditions we obtain bounds on the eigenvalues for the helmholtz equation these eigenvalue bounds are obtained as a consequence of estimates on the field of values tilde a n 2 u u l 2 tilde q n 2 u u l 2 where tilde a n 2 is the weighted collocation matrix and tilde q n 2 is the preconditioner the preconditioner tilde q n 2 is robust in the sense that it provides bounds on the h 1 0 w condition number of tilde q n 2 1 tilde l n 2 when tilde l n 2 is the weighted collocation matrix associated with the general elliptic operator lu delta u a 1u x a 2u y a 0u

bifurcation from the first eigenvalue of some nonlinear elliptic operators in banach spaces where r is a spectral parameter and a b are operators acting from a certain banach space into its dual in the papers by idogawa and otani 9 and chan et al 4 the existence of the rst variational and simple eigenvalue 1 of eq 1 1 is proved if a and b are single valued subdi erentials of certain positive convex functionals f1 and f2 respectively under some additional assumptions they proved that the problem has a positive solution if and only if 1 they provide examples of quasilinear elliptic boundary value problems on the bounded domain rn with smooth boundary in our paper we show that a slight modi cation of the assumptions in 4 allows us to extend their results also for problems in unbounded domains including the case of rn putting some additional assumptions on a and b which seems to be natural for a wide class of quasilinear equations we prove that there is a neighbourhood of 1 which does not contain any other eigenvalue than 1 finally under the assumption

eigenvalues for radially symmetric fully nonlinear singular or degenerate operators abstract in this paper we present a one dimensional and radial theory for the existence of eigenvalues and eigenfunctions for fully nonlinear elliptic 1 homogeneous operators 1 a general theory for the first eigenvalue and eigenfunction exists in the frame of viscosity solutions but in this particular case a simpler theory can be established that extends via degree theory to obtain the complete set of eigenvalues and eigenfunctions characterized by the number of zeros

preconditioning chebyshev spectral collocation by finite difference operators in 1979 orszag proposed a finite difference preconditioning of the chebyshev collocation discretization of the poisson equation in 1984 haldenwang labrosse abboudi and deville gave analytic formulae for the eigenvalues of this preconditioned operator in the one dimensional case experimental results over many years have shown the effectiveness of this procedure and appropriate bounds on the eigenvalues in two dimensions however there have been no mathematical proofs describing the behavior of the eigenvalues in two or more dimensions in this work we consider the generalized field of values u hat a nu u l nu where hat a n is the matrix of the chebyshev collocation scheme and l n is the matrix of the finite difference operator for the case of the chebyshev collocation of the helmoltz operator au delta u au a ge 0 preconditioned by the finite difference operator associated with the helmholtz operator bu delta u bu b ge 0 we prove that there are two constants 0 0 and u hat a nu u l nu le lambda 1 these results extend to higher dimensions and to bounds on l n 1 hat a n 1 w and hat a n 1 l n 1 w in the general case where au delta u a 1u x a 2u y au