Open Problems in Singular Integral Theory

Open problems in matrix theory

We survey some open problems in matrix theory by briefly describing their history and current state. 2000 Mathematics Subject Classification: 15A15, 15A18, 15A60, 15A29, 15-02, 05B20, 05C07.

Transmission problems and spectral theory for singular integral operators on Lipschitz domains

Here, ∆ is the Laplacian, μ ∈ R is a fixed parameter, ν is the outward unit normal to Ω, and Ω+ := Ω, Ω− := Rn \ Ω̄. For 1 < p < ∞, L̇p1(∂Ω) is the classical homogeneous Lp-based Sobolev spaces of order one on ∂Ω, M denotes the non-tangential maximal operator, ∂ν is the normal derivative and all restrictions to the boundary are taken in the non-tangential limit sense; detailed definitions are giv...

Nonlinear Unsteady Flow Problems by Multidimensional Singular Integral Representation Analysis

A two-dimensional nonlinear aerodynamics representation analysis is proposed for the investigation of inviscid flowfields of unsteady airfoils. Such problems are reduced to the solution of a nonlinear multidimensional singular integral equation as the source and vortex strength distributions are dependent on the history of these distributions on the NACA airfoil surface. A turbulent boundary la...

On the Spectral Theory of Singular Integral Operators.

THEOREM 4. Let a be an arbitrarily often differentiable function on the real line having countably many isolated zeros s1, S2, . . . offinite orders ml, M2, ..., respectively. Denote by A the multiplication operator mapping x into Ax = ax and let E be any linear space satisfying X = E (3 R*. Then there is a unique continuous linear op rator A-l :X -X whose range is orthogonal to E and is such t...

Algorithmic Information Theory: Open Problems