Regularity of the Maxwell equations in heterogeneous media and Lipschitz domains

This note establishes regularity estimates for the solution of the Maxwell equations in Lipschitz domains with non-smooth coefficients and minimal regularity assumptions. The argumentation relies on elliptic regularity estimates for the Poisson problem with non-smooth coefficients.     

Asymptotic behavior of the spectrum of the Maxwell operator

For the distribution functions of the positive and the negative eigenvalues of the operator in a domain with a smooth boundary, one obtains the asymptotic formula N^±()=(3²)^–1 mes ·³+0(²). Under additional assumptions on the properties of the geodesic billiard in , one shows that N±()= (3²)^–1 mes ·³+0(²).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 127, pp. 169–180, 1983.     

The stokes operator with Neumann boundary conditions in Lipschitz domains

In the first part of the paper, we give a satisfactory definition of the Stokes operator in Lipschitz domains in \mathbbRⁿ {\mathbb{R}^n} when boundary conditions of Neumann type are considered. We then proceed to establish optimal global Sobolev regularity results for vector fields in the domains of fractional powers of this Neumann–Stokes operator. Finally, we study the existence, regularity, and uniqueness of mild solutions of the Navier–Stokes system with Neumann boundary conditions. Bibliography: 43 titles. Illustrations: 2 figures.     

Estimate for the remainder in the Tate formula

Translated from Matematicheskie Zametki, Vol. 50, No. 5, pp. 149–151, November, 1991.     

Long-time asymptotics for the porous medium equation: The spectrum of the linearized operator

We compute the complete spectrum of the displacement Hessian operator, which is obtained from the confined porous medium equation by linearization around its stationary attractor, the Barenblatt profile. On a formal level, the operator is conjugate to the Hessian of the entropy via similarity transformation. We show that the displacement Hessian can be understood as a self-adjoint operator and find that its spectrum is purely discrete. The knowledge of the complete spectrum and the explicit information about the corresponding eigenfunctions give new insights on the convergence and higher order asymptotics of solutions to the porous medium equation towards its attractor. More precisely, the inspection of the eigenfunctions allows to identify symmetries in R^N${\mathbb{R}}^N$ with flows whose rates of convergence are faster than the uniform, translation-governed bound. The present work complements the analogous study of Denzler & McCann for the fast-diffusion equation.     

On the asymptotics of the spectrum of a nonsemibounded vector Sturm-Liouville operator

On the half-line, we consider a vector Sturm-Liouville operator with a potential that is unbounded below. Asymptotic formulas for the spectrum are given. These formulas involve the eigenvalues of the matrix potential as well as the “rotational velocities” of the eigenvectors.     

Reduced Weyl asymptotics for pseudodifferential operators on bounded domains I. The finite group case

Let G⊂O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn, and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A₀ in L²(Rn) with G-invariant Weyl symbol, and assume that it is semi-bounded from below. We show that the spectrum of the Friedrichs extension A of the operator is discrete, and derive asymptotics for the number Nχ(λ) of eigenvalues of A less or equal λ and with eigenfunctions in the χ-isotypic component of L²(X) as λ→∞, giving also an estimate for the remainder term in case that G is a finite group. In particular, we show that the multiplicity of each unitary irreducible representation in L²(X) is asymptotically proportional to its dimension.     

Reduced Weyl asymptotics for pseudodifferential operators on bounded domains II. The compact group case

Let G⊂O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn, and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A₀ in L²(Rn) that commutes with the regular representation of G, and assume that it is elliptic on X. We show that the spectrum of the Friedrichs extension A of the operator is discrete, and using the method of the stationary phase, we derive asymptotics for the number Nχ(λ) of eigenvalues of A equal or less than λ and with eigenfunctions in the χ-isotypic component of L²(X) as λ→∞, giving also an estimate for the remainder term for singular group actions. Since the considered critical set is a singular variety, we recur to partial desingularization in order to apply the stationary phase theorem.     

Three-term asymptotics of the discrete spectrum of the Dirichlet problem for domains with infinite perimeter

Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 28, No. 2, pp. 78–79, April–June, 1994.     

The operator remainder in the Euler-Maclaurin formula

We give a formula for the remainder term in the operator-theoretic formulation of the Euler-Maclaurin formula, using Laplace transforms. There is also a related formula for the partial sums.Keywords and phrases: Euler-Maclaurin sum formula, remainder term, Laplace transform, Bernoulli polynomial.Dedicated to Professor Janos Aczél on his 60th birthday     

Semiclassical asymptotics of the spectrum near the lower boundary of spectral clusters for a Hartree-type operator

Mathematical Notes - The eigenvalue problem for a perturbed two-dimensional resonant oscillator is considered. The exciting potential is given by a nonlocal nonlinearity of Hartree type with smooth...     

A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator

We prove a Sobolev inequality with remainder term for the imbedding , arbitrary, generalizing a corresponding result of Bianchi and Egnell for the case m = 1. We also show that the manifold of least energy solutions of the equation is a nondegenerate critical manifold for the corresponding variational integral. Finally we generalize the results of J. M. Coron on the existence of solutions of equations with critical exponent on domains with nontrivial topology to the biharmonic operator.Received: 21 March 2002, Accepted: 5 November 2002, Published online: 16 May 2003     

Estimate of the remainder term in Tate's formula

An estimate of the remainder term in Tate's formula is deduced and a property of finite order points of elliptic curves is pointed out.Translated from Matematicheskie Zametki, Vol. 3, No. 3, pp. 271–278, March, 1968.