Eugene Shargorodsky

Bernoulli Free Boundary Problems

Introduction Bernoulli free boundaries Type-

QUASI-CONFORMAL MAPPINGS AND PERIODIC SPECTRAL PROBLEMS IN DIMENSION TWO

({\mathbf I})

On the limit behaviour of second order relative spectra of self-adjoint operators

problems Proofs of main results Appendix A. Auxiliary results Bibliography Index.

A Riemann-Hilbert problem and the Bernoulli boundary condition in the variational theory of stokes waves

We study spectral properties of second-order elliptic operators with periodic coefficients in dimension two. These operators act in periodic simply-connected waveguides, with either Dirichlet, or Neumann, or the third boundary condition. The main result is the absolute continuity of the spectra of such operators. The cornerstone of the proof is an isothermal change of variables, reducing the metric to a flat one and the waveguide to a straight strip. The main technical tool is the quasiconformal variant of the Riemann mapping theorem.

Riemann–Hilbert theory for problems with vanishing coefficients that arise in nonlinear hydrodynamics

It is well known that the standard projection methods allow one to recover the whole spectrum of a bounded self-adjoint operator but they often lead to spectral pollution, i.e. to spurious eigenvalues lying in the gaps of the essential spectrum. Methods using second order relative spectra are free from this problem, but they have not been proven to approximate the whole spectrum. L. Boulton (2006, 2007) has shown that second order relative spectra approximate all isolated eigenvalues of finite multiplicity. The main result of the present paper is that second order relative spectra do not in general approximate the whole of the essential spectrum of a bounded self-adjoint operator.

LEVEL SETS OF THE RESOLVENT NORM OF A LINEAR OPERATOR REVISITED

Abstract This paper concerns the question of equivalence between the Euler–Lagrange equation of a certain functional and periodic Stokes waves on the surface of an infinitely deep irrotational incompressible flow of an ideal fluid under gravity. Of particular concern is Bernoullis constant-pressure condition on a free surface.

An Estimate for the Morse Index of a Stokes Wave

Motivated by a question from mathematical hydrodynamics, this paper studies the solution set of Riemann-Hilbert problems on the unit disc D in C of the form ˆ = a’ on @D; where ’; ˆ belong to subclasses of N + , the Nevanlinna-Smirnov functions on D, and the coecient a is a real-valued non-negative function which vanishes at points of @D.

Applications of Blaschke Products to the Spectral Theory of Toeplitz Operators

It is proved that the resolvent norm of an operator with a compact resolvent on a Banach space

An algebra of integral operators with fixed singularities in kernels

X

More on the density of analytic polynomials in abstract Hardy spaces

cannot be constant on an open set if the underlying space or its dual is complex strictly convex. It is also shown that this is not the case for an arbitrary Banach space: there exists a separable, reflexive space