Shape sensitivity for the Laplace-Beltrami operator with singularities

This paper considers shape sensitivity analysis for the Laplace-Beltrami operator formulated on a two-dimensional manifold with a fracture. We characterize the shape gradient of a functional as a bounded measure on the manifold and decompose it into a “distributed gradient” supported on the manifold, plus a singular part that we derive as the limit of a “jump” through the crack and Dirac measures at the crack extremities. The important point is that we introduce a technique that is not dimension dependent, and makes no use of classical arguments such as the maximum principle or continuation uniqueness. The technique makes use of a family of envelopes surrounding the fracture which enable us to relax certain terms and to overcome the lack of regularity resulting from the presence of the fracture. We use the min-max differentiation in order to avoid taking the derivative of the state equation and to manage the crack's singularities. Therefore, we write the functional in a min-max formulation on a space which takes into account the hidden boundary regularity established by the tangential extractor method.     

Applications of a Laplace-Beltrami operator for Jack polynomials

An explicit computation is made for a Laplace-Beltrami type operator for Jack polynomials. As applications we obtain: combinatorial formula, determinantal formula and raising operator formula for Jack polynomials, as well as an iterative formula for the Littlewood-Richardson coefficients. One special case of our results implies Mimachi-Yamada’s result on Jack polynomials of rectangular shapes.     

Approximation of functions with zero integrals over balls by linear combinations of the Laplace operator eigenfunctions

It is proved that the special linear combinations of Bessel functions are dense in the C ^∞-topology in the space of functions with zero integrals over balls of fixed radii on an arbitrary open domain U ì \mathbbRⁿ U \subset {\mathbb{R}^n} . Some generalizations of this result for solutions of some convolution equations of the form f * T = 0, where T is radial, are obtained. Analogous results for rank-one symmetric spaces of the noncompact type are considered.     

Inequalities and separation for the Laplace-Beltrami differential operator in Hilbert spaces

In this paper we have studied the separation for the Laplace-Beltrami differential operator of the form

     

Formula for the first regularized trace for the Laplace-Beltrami operator with a potential

The first regularized trace of the Laplace-Beltrami operator with an odd potential on two-dimensional sphere is computed. Bibliography: 4 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 18, pp. 79–105. 1995.     

Outlines for the computation of the multiplicities of the spectra of orthogonal sums

If A and B are operators in the spaces X and Y, respectively, and if the operator B has many sets , , such that the manifolds p is a polynomial are dense in the space Y, then Here a=(the multiplicity of the spectrum of the operator A)=mindimL: span (AⁿL:n0)}=X. For example, if B=T^–g is a Toeplitz operator in the space H² with antianalytic symbol) and if g (the polynomial convex hull of the spectrum (A)) , then. Conversely, if and, then (under some assumptions on the regularity of the function f we have. One also gives examples of univalent and essentially univalent functions f (f H), for which Tf>1.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 150–158, 1983.     

Estimates for eigenvalues of the Paneitz operator

For an n  -dimensional compact submanifold Mⁿ$M^n$ in the Euclidean space R^N${\mathbf{R}}^N$, we study estimates for eigenvalues of the Paneitz operator on Mⁿ$M^n$. Our estimates for eigenvalues are sharp.     

Multiplicities of singularities of eigenfunctions for the Laplace—Beltrami operator

Institute for Problems in Information Transmission, Russian Academy of Sciences. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 29, No. 1, pp. 80–82, January–March, 1995.     

Estimates for the Hill operator, II

Consider the Hill operator Ty=-y^″+q(t)y in L²(R), where the real potential q is 1-periodic and q,q^′∈L²(0,1). The spectrum of T consists of spectral bands separated by gaps γ_n,n?1 with length |γ_n|?0. We obtain two-sided estimates of the gap lengths ∑n²|γ_n|² in terms of . Moreover, we obtain the similar two-sided estimates for spectral data (the height of the corresponding slit on the quasimomentum domain, action variables for the KdV equation and so on). In order prove this result we use the analysis of a conformal mapping corresponding to quasimomentum of the Hill operator. That makes it possible to reformulate the problems for the differential operator as the problems of the conformal mapping theory. Then the proof is based on the analysis of the conformal mapping, the embedding theorems and the identities. Furthermore, we obtain the similar two-sided estimates for potentials which have p?2 derivatives.     

Estimates for character sums in number fields

We obtain asymptotic formulae for the moments of coefficients of Artin-WeilL-functions (defined by equation (1)). As a by-product of our investigations, we strengthen and generalise the results of several authors cited in the references.