the proper dissipative extensions of a dual pair let a and widetilde a be dissipative operators on a hilbert space mathcal h and let a widetilde a form a dual pair i e a subset widetilde a resp widetilde a subset a we present a method of determining the proper dissipative extensions widehat a of this dual pair i e a subset widehat a subset widetilde a provided that mathcal d a cap mathcal d widetilde a is dense in mathcal h applications to symmetric operators symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators are discussed finally we investigate the stability of the numerical range of the different dissipative extensions

regularity of inverses of sobolev deformations with finite surface energy abstract let u be a sobolev w 1 p map from a bounded open set r n to r n we assume u to satisfy some invertibility properties that are natural in the context of nonlinear elasticity namely the topological condition inv and the orientation preserving constraint det d u 0 these deformations may present cavitation which is the phenomenon of void formation we also assume that the surface created by the cavitation process has finite area if p n 1 we show that a suitable defined inverse of u is a sobolev map a partial result is also given for the critical case p n 1 the proof relies on the techniques used in the study of cavitation

on inverse spectral theory for self adjoint extensions mixed types of spectra abstract lethbe a symmetric operator in a separable hilbert space h suppose thathhas some gapj we shall investigate the question about what spectral properties the self adjoint extensions ofhcan have inside the gapjand provide methods on how to construct self adjoint extensions ofhwith prescribed spectral properties insidej under some weak assumptions about the operatorhwhich are satisfied e g provided the deficiency indices ofhare infinite and the operator h 1is compact for one regular point ofh we shall show that for every auxiliary self adjoint operatorm in the hilbert space h and every open subsetj0of the gapjofhthere exists a self adjoint extensionhofhsuch that insidejthe self adjoint extensionhofhhas the same absolutely continuous and the same point spectrum as the given operatorm and the singular continuous spectrum ofhinjequals the closure ofj0inj moreover we shall present a method of how to construct such a self adjoint extensionh via our methods it is possible to construct new kinds of self adjoint realizations of the laplacian on a bounded domain in r d d 1 with spectral properties very different from the spectral properties of the self adjoint realizations known before mathematics subject classification 1991 47a10 47a60 47b25 47e05 47f05

critical properties of random site percolation in two and three dimensions a monte carlo study abstract we measure the critical exponents of two dimensional and three dimensional random site percolation and find excellent agreement with nienhuis exact results two dimensions and good agreement with other numerical work three dimensions we also measure the correlation length amplitude ratio and the mean cluster size amplitude ratio in two and three dimensions we report values of 4 0 0 5 2 d 2 0 0 5 3 d and 75 40 26 2 d and 8 3 2 3 d for the correlation length and the mean cluster size amplitude ratios respectively a direct measurement is made of the fractal dimension d f at the critical point p c 0 50 of the triangular lattice d f is 1 90 0 01 while at p c 0 3117 for the simple cubic lattice d f is 2 50 0 02

on the negative squares of indefinite sturm liouville operators abstract the number of negative squares of all self adjoint extensions of a simple symmetric operator of defect one with finitely many negative squares in a krein space is characterized in terms of the behaviour of an abstract titchmarsh weyl function near 0 and these results are applied to a large class of symmetric and self adjoint indefinite sturm liouville operators with indefinite weight functions

the similarity problem for j nonnegative sturm liouville operators abstract sufficient conditions for the similarity of the operator a 1 r x d 2 d x 2 q x with an indefinite weight r x sgn x r x are obtained these conditions are formulated in terms of titchmarsh weyl m coefficients sufficient conditions for the regularity of the critical points 0 and of j nonnegative sturm liouville operators are also obtained this result is exploited to prove the regularity of 0 for various classes of sturm liouville operators this implies the similarity of the considered operators to self adjoint ones in particular in the case r x sgn x and q l 1 r 1 x d x we prove that a is similar to a self adjoint operator if and only if a is j nonnegative the latter condition on q is sharp i e we construct q 1 l 1 r 1 x d x such that a is j nonnegative with the singular critical point 0 hence a is not similar to a self adjoint operator for periodic and infinite zone potentials we show that j positivity is sufficient for the similarity of a to a self adjoint operator in the case q 0 we prove the regularity of the critical point 0 for a wide class of weights r this yields new results for forward backward diffusion equations

a minimization problem involving variation of the domain keywords calculus of variations existence and regularity of minimizers deformation of cristals energy functional reference caa article 1992 003doi 10 1002 cpa 3160450704view record in web of science record created on 2008 11 25 modified on 2017 05 12

invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity in this paper we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture the main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface accordingly we define a functional that measures the area of the created surface this functional has relationships with the theory of cartesian currents we show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient and that the weak limit of one to one almost everywhere deformations is also one to one almost everywhere we then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy taking into account orientation preserving and non interpenetration conditions

scattering matrices and weyl functions for a scattering system a a 0 consisting of self adjoint extensions a and a 0 of a symmetric operator a with finite deficiency indices the scattering matrix s and a spectral shift function are calculated in terms of the weyl function associated with a boundary triplet for a and a simple proof of the krein birman formula is given the results are applied to singular sturm liouville operators with scalar and matrix potentials to dirac operators and to schrodinger operators with point interactions

the fractal dimension of the minimum path in two and three dimensional percolation the authors calculate the fractal dimension d min of the shortest path l between two points on a percolation cluster where l approximately rd min and r is the pythagorean distance between the points they find d min 1 130 or 0 002 for d 2 and 1 34 or 0 01 for d 3

symmetric operators with real defect subspaces of the maximal dimension applications to differential operators abstract let h be a hilbert space and let a be a simple symmetric operator in h with equal deficiency indices d n a we show that if for all in an open interval i r the dimension of defect subspaces n a ker a coincides with d then every self adjoint extension a a has no continuous spectrum in i and the point spectrum of a is nowhere dense in i application of this statement to differential operators makes it possible to generalize the known results by weidmann to the case of an ordinary differential expression with both singular endpoints and arbitrary equal deficiency indices of the minimal operator

spectral theory of schrodinger operators with infinitely many point interactions and radial positive definite functions abstract a number of results on radial positive definite functions on r n related to schoenberg s integral representation theorem are obtained they are applied to the study of spectral properties of self adjoint realizations of two and three dimensional schrodinger operators with countably many point interactions in particular we find conditions on the configuration of point interactions such that any self adjoint realization has purely absolutely continuous non negative spectrum we also apply some results on schrodinger operators to obtain new results on completely monotone functions

regularity properties of deformations with finite energy the purpose of this paper is to point out some regularity properties of a class of functions which play an important role in nonlinear elasticity

trace formulas for additive and non additive perturbations abstract trace formulas for pairs of self adjoint maximal dissipative and accumulative as well as other types of resolvent comparable operators are obtained in particular the existence of a complex valued spectral shift function for a pair h h of maximal accumulative operators has been proved we investigate also the existence of a real valued spectral shift function moreover we treat in detail the case of additive trace class perturbations assuming that h and h h v are maximal accumulative and v is trace class we prove the existence of a summable complex valued spectral shift function we also obtain trace formulas for pairs h h assuming only that h and h are resolvent comparable in this case the determinant of the characteristic function of h is involved in trace formulas the results rely on the technique of boundary triplets for densely and non densely defined closed symmetric operators the approach allows to express the spectral shift function of a pair of extensions in terms of the abstract weyl function and the boundary operator we improve and generalize certain classical results of m g kre n for pairs of self adjoint and dissipative operators results of a v rybkin for such pairs as well as of v m adamjan b s pavlov and m g kre n for pairs h h with a maximal dissipative operator h

on the spectral theory of singular indefinite sturm liouville operators we consider a singular sturm liouville differential expression with an indefinite weight function and we show that the corresponding self adjoint differential operator in a krein space locally has the same spectral properties as a definitizable operator

continuous spectrum and square integrable solutions of differential operators with intermediate deficiency index we explore the connection between square integrable solutions for real values of the spectral parameter and the continuous spectrum of self adjoint ordinary differential operators with arbitrary deficiency index d we show that if for all in an open interval i there are d of linearly independent square integrable solutions then for every extension of dmin the point spectrum is nowhere dense in i and there is a self adjoint extension of smin which has no continuous spectrum in i this analysis is based on our construction of limit point lp and limit circle lc solutions obtained recently in an earlier paper

canonical systems of differential equations with self adjoint interface conditions on graphs for n canonical systems of differential equations the corresponding n copies of their domain 0 are thought of as a graph with vertex 0 an interface condition at 0 is given by a so called nevanlinna pair explicit formulae are deduced for the spectral representation of the corresponding underlying self adjoint relation and the generalized fourier transformation furthermore results on compressions of the fourier transformation to closed linear subspaces and the multiplicity of the eigenvalues if the spectrum is discrete are presented

test of universal finite size scaling in two dimensional site percolation traditional two scale factor universality concerning the number of clusters within a correlation volume near the percolation threshold is reconfirmed by a comparison of site percolation on square and triangular lattices we also test universality at the threshold e g of the ratio where is the correlation length and l the lattice size at the threshold the result is sensitive to the system s shape and aspect ratio boundary conditions the algorithm the detailed units measuring and l and possibly other factors as yet unexplored

shorting parallel addition and form sums of nonnegative selfadjoint linear relations abstract we extend the operations of shorting and parallel addition from the cone of bounded nonnegative selfadjoint operators in a hilbert space to the set of all nonnegative selfadjoint linear relations new properties of these operations and connections with the cayley transforms are established it is shown that for a pair of nonnegative selfadjoint linear relations there exist in general two arithmetic harmonic means applications of the arithmetic harmonic and arithmetic harmonic means to the theory of nonnegative selfadjoint extensions of nonnegative symmetric linear relations are given

boundary triples and weyl m functions for powers of the jacobi differential operator abstract the abstract theory of boundary triples is applied to the classical jacobi differential operator and its powers in order to obtain the weyl m function for several self adjoint extensions with interesting boundary conditions separated periodic and those that yield the friedrichs extension these matrix valued nevanlinna herglotz m functions are to the best knowledge of the author the first explicit examples to stem from singular higher order differential equations the creation of the boundary triples involves taking pieces determined in 26 of the principal and non principal solutions of the differential equation and putting them into the sesquilinear form to yield maps from the maximal domain to the boundary space these maps act like quasi derivatives which are usually not well defined for all functions in the maximal domain of singular expressions however well defined regularizations of quasi derivatives are produced by putting the pieces of the non principal solutions through a modified gram schmidt process

a remark on discrete wave scattering we consider the spectral problem for the adjacency matrix of a discrete star graph composed of a compact cluster and a few semi infinite periodic leads attached based on the spectral properties of the adjacency matrix we develop the lax phillips scattering theory for the corresponding discrete wave equation

generalized q functions and dirichlet to neumann maps for elliptic differential operators the classical concept of q functions associated to symmetric and selfadjoint operators due to m g krein and h langer is extended in such a way that the dirichletto neumann map in the theory of elliptic differential equations can be interpreted as a generalized q function for couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit krein type formulas for the difference of the resolvents and trace formulas in an h 2 framework are obtained

on symmetries in the theory of finite rank singular perturbations for a nonnegative self adjoint operator a0 acting on a hilbert space h singular perturbations of the form a0 v v 1nbij j i are studied under some additional requirements of symmetry imposed on the initial operator a0 and the singular elements j a concept of symmetry is defined by means of a one parameter family of unitary operators u that is motivated by results due to r s phillips the abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials v and the corresponding self adjoint realizations of a0 v the results are applied for the investigation of singular perturbations of the schrodinger operator in l2 r3 and for the study of a fractional p adic schrodinger type operator with point interactions

percolation and cluster structure parameters the radius of gyration the squared radius of gyration of percolation clusters are determined in terms of the clusters second spatial moments using the enhanced hoshen kopelman algorithm for square and triangular lattices containing sites correlation length exponents and related exponents as well as their corresponding amplitudes are calculated in terms of the squared radius of gyration above and below the percolation threshold a coefficient for cluster compactness that is based on the squared radius of gyration of a cluster is introduced that coefficient is compared with a similar coefficient of compactness that is based on the cluster cyclomatic number

a krein like formula for singular perturbations of self adjoint operators and applications abstract given a self adjoint operator a d a h h and a continuous linear operator d a x with range h 0 x a banach space we explicitly construct a family a of self adjoint operators such that any a coincides with the original a on the kernel of such a family is obtained by giving a krein like formula where the role of the deficiency spaces is played by the dual pair x x the parameter belongs to the space of symmetric operators from x to x when x c one recovers the h 2 construction of kiselev and simon and so to some extent our results can be regarded as an extension of it to the infinite rank case considering the situation in which h l 2 r n and is the trace restriction operator along some null subset we give various applications to singular perturbations of non necessarily elliptic pseudo differential operators thus unifying and extending previously known results

fracture surfaces and the regularity of inverses for bv deformations motivated by nonlinear elasticity theory we study deformations that are approximately differentiable orientation preserving and one to one almost everywhere and in addition have finite surface energy this surface energy mathcal e was used by the authors in a previous paper and has connections with the theory of currents in the present paper we prove that mathcal e measures exactly the area of the surface created by the deformation this is done through a proper definition of created surface which is related to the set of discontinuity points of the inverse of the deformation in doing so we also obtain an sbv regularity result for the inverse

the functional model for maximal dissipative operators translation form an approach in the spirit of operator knots in this article we develop a functional model for a general maximal dissipative operator we construct the selfadjoint dilation of such operators unlike previous functional models our model is given explicitly in terms of parameters of the original operator making it more useful in concrete applications for our construction we introduce an abstract framework for working with a maximal dissipative operator and its anti dissipative adjoint and make use of the straus characteristic function in our setting explicit formulae are given for the selfadjoint dilation its resolvent a core and the completely non selfadjoint subspace minimality of the dilation is shown the abstract theory is illustrated by the example of a schrodinger operator on a half line with dissipative potential and boundary condition and connections to existing theory are discussed

inverse spectral theory for symmetric operators with several gaps scalar type weyl functions let s be the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non zero deficiency indices let j be an open subset of r if there exists a self adjoint extension s0 of s such that j is contained in the resolvent set of s0 and the associated weyl function of the pair s s0 is monotone with respect to j then for any self adjoint operator r there exists a self adjoint extension s such that the spectral parts s j and rj are unitarily equivalent it is shown that for any extension s of s the absolutely continuous spectrum of s0 is contained in that one of s moreover for a wide class of extensions the absolutely continuous parts of s and s are even unitarily equivalent

on the many dirichlet laplacians on a non convex polygon and their approximations by point interactions abstract by birman and skvortsov it is known that if is a planar curvilinear polygon with n non convex corners then the laplace operator with domain h 2 h 0 1 is a closed symmetric operator with deficiency indices n n here we provide a kre n type resolvent formula for any self adjoint extensions of such an operator i e for the set of self adjoint non friedrichs dirichlet laplacians on and show that any element in this set is the norm resolvent limit of a suitable sequence of friedrichs dirichlet laplacians with n point interactions

a remark on schatten von neumann properties of resolvent differences of generalized robin laplacians on bounded domains in this note we investigate the asymptotic behavior of the s numbers of the resolvent difference of two generalized self adjoint maximal dissipative or maximal accumulative robin laplacians on a bounded domain with smooth boundary for this we apply the recently introduced abstract notion of quasi boundary triples and weyl functions from extension theory of symmetric operators together with krein type resolvent formulae and well known eigenvalue asymptotics of the laplace beltrami operator on it is shown that the resolvent difference of two generalized robin laplacians belongs to the schatten von neumann class of any order p for which

on kreĭn s formula abstract kre n s formula provides a parametrization of the generalized resolvents and straus extensions of a closed symmetric operator with equal possibly infinite defect numbers in a hilbert space in terms of nevanlinna families in a parameter space the aim of this note is to give a simple complete analytical proof of kre n s formula

sharp eigenvalue estimates for rank one perturbations of nonnegative operators in krein spaces abstract let a and b be selfadjoint operators in a krein space and assume that the resolvent difference of a and b is of rank one in the case that a is nonnegative and i is an open interval such that a i consists of isolated eigenvalues we prove sharp estimates on the number and multiplicities of eigenvalues of b in i the general result is illustrated with eigenvalue estimates for singular indefinite sturm liouville problems

on eisenbud s and wigner s r matrix a general approach abstract the main objective of this paper is to give a rigorous treatment of wigner s and eisenbud s r matrix method for scattering matrices of scattering systems consisting of two selfadjoint extensions of the same symmetric operator with finite deficiency indices in the framework of boundary triplets and associated weyl functions an abstract generalization of the r matrix method is developed and the results are applied to schrodinger operators on the real axis

generalized resolvents and the boundary value problems for hermitian operators with gaps a hermitian operator a with gaps j j 1 j m is studied the self adjoint extensions which put exactly kj eigenvalues into each gap j j in particular for kj 0 1 j m the extensions preserving the gaps are described in terms of boundary conditions the generalized resolvents of the extensions with the indicated properties are described also a solvability criterion and description of all the solutions of the hamburger moment problem with supports in r j 1m j j are obtained in terms of the nevanlinna matrix

boundary value problems for elliptic partial differential operators on bounded domains for a symmetric operator or relation a with infinite deficiency indices in a hilbert space we develop an abstract framework for the description of symmetric and self adjoint extensions a of a as restrictions of an operator or relation t which is a core of the adjoint a this concept is applied to second order elliptic partial differential operators on smooth bounded domains and a class of elliptic problems with eigenvalue dependent boundary conditions is investigated

boundary relations and their weyl families the concepts of boundary relations and the corresponding weyl families are introduced let s be a closed symmetric linear operator or more generally a closed symmetric relation in a hilbert space h let h be an auxiliary hilbert space let graphics and let jh be defined analogously a unitary relation g from the krein space h 2 j h to the kre in space h 2 j h is called a boundary relation for the adjoint s if ker gamma s the corresponding weyl family m lambda is de ned as the family of images of the defect subspaces n over cap lambda lambda is an element of c r under gamma here gamma need not be surjective and is even allowed to be multi valued while this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex hilbert space h and the class of unitary relations gamma h 2 j h h 2 j h it also generalizes the notion of so called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems moreover these new notions yield for instance the following realization theorem every h valued maximal dissipative for lambda is an element of c holomorphic family of linear relations is the weyl family of a boundary relation which is unique up to unitary equivalence if certain minimality conditions are satisfied further connections between analytic and spectral theoretical properties of weyl families and geometric properties of boundary relations are investigated and some applications are given

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

spectral analysis of selfadjoint elliptic differential operators dirichlet to neumann maps and abstract weyl functions abstract the spectrum of a selfadjoint second order elliptic differential operator in l 2 r n is described in terms of the limiting behavior of dirichlet to neumann maps which arise in a multi dimensional glazman decomposition and correspond to an interior and an exterior boundary value problem this leads to pde analogs of renowned facts in spectral theory of odes the main results in this paper are first derived in the more abstract context of extension theory of symmetric operators and corresponding weyl functions and are applied to the pde setting afterwards

a bound for the eigenvalue counting function for krein von neumann and friedrichs extensions for an arbitrary open nonempty bounded set rn n n and sufficiently smooth coefficients a b q we consider the closed strictly positive higher order differential operator a 2m a b q in l2 defined on w02m 2 associated with the differential expression 2m a b q j k 1n i j bj aj k i k bk q m m n and its krein von neumann extension ak 2m a b q in l2 denoting by n ak 2m a b q 0 the eigenvalue counting function corresponding to the strictly positive eigenvalues of ak 2m a b q we derive the bound n ak 2m a b q cvn 2 n 1 2m2m n n 2m n 2m 0 where c c a b q 0 with c in 0 0 is connected to the eigenfunction expansion of the self adjoint operator a 2m a b q in l2 rn defined on w2m 2 rn corresponding to 2m a b q here vn n 2 n 2 2 denotes the euclidean volume of the unit ball in rn our method of proof relies on variational considerations exploiting the fundamental link between the krein von neumann extension and an underlying abstract buckling problem and on the distorted fourier transform defined in terms of the eigenfunction transform of a 2 a b q in l2 rn we also consider the analogous bound for the eigenvalue counting function for the friedrichs extension af 2m a b q in l2 of a 2m a b q no assumptions on the boundary of are made

on essential spectra of singular linear hamiltonian systems abstract the paper is concerned with essential spectra of singular linear hamiltonian systems of arbitrary order with arbitrary equal defect indices several sufficient conditions for the essential spectral points are given in terms of the number of linearly independent square integrable solutions of the corresponding hamiltonian system and a sufficient and necessary condition for the essential spectral points is obtained for hamiltonian systems of even order with one singular endpoint an advantage of these results is that one can determine the essential spectral points of hamiltonian systems by the information of solutions obtained by numerous tools available in the fundamental theory of differential equations in addition two illustrative examples are provided to show how to get some information about the essential spectrum by our results

orlicz sobolev nematic elastomers abstract we extend the existence theorems in barchiesi et al 2017 for models of nematic elastomers and magnetoelasticity to a larger class in the scale of orlicz spaces these models consider both an elastic term where a polyconvex energy density is composed with an unknown state variable defined in the deformed configuration and a functional corresponding to the nematic energy or the exchange and magnetostatic energies in magnetoelasticity where the energy density is integrated over the deformed configuration in order to obtain the desired compactness and lower semicontinuity we show that the regularity requirement that maps create no new surface can still be imposed when the gradients are in an orlicz class with an integrability just above the space dimension minus one we prove that the fine properties of orientation preserving maps satisfying that regularity requirement namely being weakly 1 pseudomonotone h 1 continuous a e differentiable and a e locally invertible are still valid in the orlicz sobolev setting

a novel method of calculating amplitude ratios by series expansions the author introduces a new method of calculating critical amplitude ratios using series which is both simple and powerful this method which gives estimates for the amplitude ratios that are neither biased by the values of the critical points nor by the critical exponents is applied to several models it is shown that this method produces results where no reliable estimates from series expansion exist in particular one finds 0 025 or 0 001 for at b2 for the 3d ising model and 220 or 10 for c c for the two dimensional percolation model in agreement with and with more accuracy than values obtained by other methods

an exact universal amplitude ratio for percolation the universal amplitude ratio for percolation in two dimensions is determined exactly using results for the dilute a model in regime 1 by way of a relationship with the q state potts model with q 4

fractal properties of the percolating backbone in three dimensions a recursive algorithm to determine the backbone of the two dimensional percolating cluster is adjusted to measure the fractal properties of the backbone of the three dimensional percolating cluster the specific properties measured are the fractal dimensionality of the backbone dbbf 1 75 or 0 04 the chemical dimensionality dt 1 26 or 0 03 and the minimum path dimensionality dmin 1 39 or 0 03

products of generalized nevanlinna functions with symmetric rational functions abstract new classes of generalized nevanlinna functions which under multiplication with an arbitrary fixed symmetric rational function remain generalized nevanlinna functions are introduced characterizations for these classes of functions are established by connecting the canonical factorizations of the product function and the original generalized nevanlinna function in a constructive manner also a detailed functional analytic treatment of these classes of functions is carried out by investigating the connection between the realizations of the product function and the original function the operator theoretic treatment of these realizations is based on the notions of rigged spaces boundary triplets and associated weyl functions

absolutely continuous functions of several variables abstract a class of absolutely continuous functions ofnvariables is introduced the absolute continuity implies continuity weak differentiability with gradient inln differentiability almost everywhere area coarea and degree formulae this gives a unified approach to some previously known conditions for the mentioned results based on monotonicity finite dilatation or higher integrability of the gradient

completion extension factorization and lifting of operators the well known results of m g kre n concerning the description of selfadjoint contractive extensions of a hermitian contraction t 1 and the characterization of all nonnegative selfadjoint extensions widetilde a of a nonnegative operator a via the inequalities a k le widetilde a le a f where a k and a f are the kre n von neumann extension and the friedrichs extension of a are generalized to the situation where widetilde a is allowed to have a fixed number of negative eigenvalues these generalizations are shown to be possible under a certain minimality condition on the negative index of the operators i t 1 t 1 and a respectively these conditions are automatically satisfied if t 1 is contractive or a is nonnegative respectively the approach developed in this paper starts by establishing first a generalization of an old result due to yu l shmul yan on completions of nonnegative block operators the extension of this fundamental result allows us to prove analogs of the above mentioned results of m g kre n and in addition to solve some related lifting problems for j contractive operators in hilbert pontryagin and kre n spaces in a simple manner also some new factorization results are derived for instance a generalization of the well known douglas factorization of hilbert space operators in the final steps of the treatment some very recent results concerning inequalities between semibounded selfadjoint relations and their inverses turn out to be central in order to treat the ordering of non contractive selfadjoint operators under cayley transforms properly

an existence theory for nonlinear elasticity that allows for cavitation in this paper the existence of minimizers in nonlinear elasticity is established under assumptions on the stored energy that permit the formation of new holes in the body such cavities have been observed in experiments on elastomers and a mathematical theory for radially symmetric cavities has been developed by ball here the full three dimensional problem is considered and an additional physically motivated energy term that is proportional to the area of the boundary of the deformed body is included the minimizers lie in a subclass of those maps in w1 p 2 p 3 that are one to one almost everywhere and preserve orientation roughly speaking this subclass consists of those maps in which cavities in one part of the body are not filled by material from other parts of the body such maps are shown to be much more regular than expected in particular some ideas of sverak are used to show that each map in this subclass has a representative which is continuous outside a set of hausdorff dimension 3 p and that this representative also satisfies lusin s condition n i e it maps lebesgue null sets onto such sets it is also shown that the distributional jacobian of such a map is a measure which is the sum of a measure that is absolutely continuous with respect to lebesgue measure and at most a countable number of dirac measures

novel domain growth law in random magnets abstract we investigate the kinetics of domain growth in ising magnets where a fraction 1 p of the magnetic atoms or ions spins are randomly substituted by non magnetic impurities we argue that close to the percolation threshold p c the statistical self similarity of the underlying structure gives rise to a novel crossover in the growth law we propose a method to detect any evidence of this new prediction from the kinetics of domain growth in the dilute ising model dim during intermediate time scales by carrying out monte carlo simulations not at p p c but at slightly higher spin concentrations we analyze the results of our extensive monte carlo simulation of the strongly diluted two dimensional ising model and find the growth to be consistent with the proposed scenario we also compare our observations with those in the recent experiments on the kinetics of ordering in rb 2 co p mg 1 p f 4

dirichlet to neumann maps abstract weyl titchmarsh m functions and a generalized index of unbounded meromorphic operator valued functions abstract we introduce a generalized index for certain meromorphic unbounded operator valued functions the class of functions is chosen such that energy parameter dependent dirichlet to neumann maps associated to uniformly elliptic partial differential operators particularly non self adjoint schrodinger operators on bounded lipschitz domains and abstract operator valued weyl titchmarsh m functions and donoghue type m functions corresponding to closed extensions of symmetric operators belong to it the principal purpose of this paper is to prove index formulas that relate the difference of the algebraic multiplicities of the discrete eigenvalues of robin realizations of non self adjoint schrodinger operators and more abstract pairs of closed operators in hilbert spaces with the generalized index of the corresponding energy dependent dirichlet to neumann maps and abstract weyl titchmarsh m functions respectively

p adic schrodinger type operator with point interactions a p adic schrodinger type operator d vy is studied d 0 is the operator of fractional differentiation and vy i j 1nbij xj xi bij c is a singular potential containing the dirac delta functions x concentrated on a set of points y x1 xn of the field of p adic numbers qp it is shown that such a problem is well posed for 1 2 and the singular perturbation vy is form bounded for 1 in the latter case the spectral analysis of self adjoint operator realizations of d vy in l2 qp is carried out

spherical schrodinger operators with δ type interactions we investigate spectral properties of spherical schrodinger operators also known as bessel operators with point interactions concentrated on a discrete set we obtain necessary and sufficient conditions for these hamiltonians to be self adjoint lower semibounded and also we investigate their spectra we also extend the classical bargmann estimate to such hamiltonians in certain cases we express the number of negative eigenvalues explicitly by means of point interactions and the corresponding intensities we apply our results to schrodinger operators in l2 rn with a singular interaction supported by an infinite family of concentric spheres

local invertibility in sobolev spaces with applications to nematic elastomers and magnetoelasticity we define a class of deformations in w 1 p omega mathbb r n p n 1 with a positive jacobian that do not exhibit cavitation we characterize that class in terms of the non negativity of the topological degree and the equality det det that the distributional determinant coincides with the pointwise determinant of the gradient maps in this class are shown to satisfy a property of weak monotonicity and as a consequence they enjoy an extra degree of regularity we also prove that these deformations are locally invertible moreover the neighbourhood of invertibility is stable along a weak convergent sequence in w 1 p and the sequence of local inverses converges to the local inverse we use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps we apply those results to prove the existence of minimizers in some models for nematic elastomers and magnetoelasticity

sturm liouville boundary value problems with operator potentials and unitary equivalence consider the minimal sturm liouville operator a amin generated by the differential expression a d2dt2 t in the hilbert space l2 r h where t t 0 in h we investigate the absolutely continuous parts of different self adjoint realizations of a in particular we show that dirichlet and neumann realizations ad and an are absolutely continuous and unitary equivalent to each other and to the absolutely continuous part of the krein realization moreover if inf ess t inf t then the part a acea ad of any self adjoint realization a of a is unitarily equivalent to ad in addition we prove that the absolutely continuous part a ac of any realization a is unitarily equivalent to ad provided that the resolvent difference a i 1 ad i 1 is compact the abstract results are applied to elliptic differential expressions in the half space

rank one perturbations in a pontryagin space with one negative square let n 1 denote the class of generalized nevanfinna functions with one negative square and let n 1 n 0 be the subclass of functions q z is an element of n with the additional properties lim y infinity q i y y 0 and lim sup y infinity y im q i y i these classes form an analytic framework for studying generalized rank one perturbations a a in a pontryagin space setting many functions appearing in quantum mechanical models of point interactions either belong to the subclass n 1 n 0 or can be associated with the corresponding generalized friedrichs extension in this paper a spectral theoretical analysis of the perturbations a and the associated friedrichs extension is carried out many results such as the explicit characterizations for the critical eigenvalues of the perturbations a are based on a recent factorization result for generalized nevanlinna functions c 2002 elsevier science usa