Annals of Communications in Mathematics

AbstractLet D be a connected bounded domain in R^2, S be its boundary which is closed, connected, and smooth, or S=(-∞,∞). Let Φ(z) be the function defined as Φ(z)=1/(2πi) ∫S(f(s)ds)/(s-z), where f∈L^1(S) and z=x+iy. The singular integral operator Af is defined as Af: =1/(iπ) ∫S(f(s)ds)/(s-t), where t∈S. This new definition simplifies the proof of the existence of Φ(t). Necessary and sufficient conditions are given for f∈L^1(S) to be the boundary value of an analytic function in D. The Sokhotsky-Plemelj formulas are derived for f∈L^1(S). Our new definition allows one to treat singular boundary values of analytic functions.

AbstractLet D be a connected bounded domain in Rn, n ≥ 2, S be its boundary, which is closed and smooth. Consider the Dirichlet problem ∆u = 0 in D, u|S = f, where f ∈ L1 (S) or f ∈ H−ℓ , where H−ℓ is the dual space to the Sobolev space Hℓ := Hℓ (S), ℓ ≥ 0 is arbitrary. The aim of this paper is to prove that the above problem has a solution for an arbitrary f ∈ L1 (S) and this solution is unique and to prove similar result for rough (distributional) boundary values. These results are new. The method of its proof, based on the potential theory, is also new. Definition of the L1 (S)-boundary value and a distributional boundary value of a harmonic in D function is given. For f ∈ L1 (S) the difficulty comes from the fact that the product of an L1 (S) function times the kernel of the potential on S is not absolutely integrable. We prove that an arbitrary f ∈ H−ℓ , ℓ > 0, can be the boundary value of a harmonic function in D.

AbstractFor a wide class of infinite boundaries and the zero boundary condition, a simple proof is given for the absence of the positive eigenvalues of the Laplacian. The objective of this work is to prove Theorem 1 in which such conditions are formulated.

AbstractA new proof is given for the uniqueness theorem for inverse obstacle scattering with non-overdetermined scattering data. It is proved that the knowledge of the scattring amplitude for a fixed wave number, fixed direction of the incident field and all directions of the scattered field in an arbitrary small cone determine the boundary of the obstacle uniquely for the Dirichle boundary condition on the obstacle.

AbstractThe Riemann problem is stated as follows: find an analytic in a domain D+ ∪ D− function Φ(z) such that (∗) Φ+(t) = G(t)Φ−(t) + g(t), t ∈ S. Here S is the boundary of D+, D− complements the complex plane to D+ ∪ S, the functions G = G(t) and g = g(t) belong to Hμ(S), the space of H¨older-continuous functions. The theory of problem (*) is developed also for continuous G. If G = 1, S ∈ C∞ and g is a tempered distribution, then problem (∗) has a solution in tempered distributions. It is proved that problem (∗) for G ∈ Lp(S) and g a tempered distribution does not make sense. It is proved that if G ∈ C∞(S), G̸ = 0 on S, and g is a distribution of the class D′, then the Riemann problem makes sense and a method for solving this problem is given. It is proved that if S = R = (−∞, ∞) and G ∈ Lp(R), where p ≥ 1 is a fixed number, then | ln G| does not belong to Lq (R) for any q ≥ 1.