Poincaré’s Variational Problem in Potential Theory

Spectral bounds for the Neumann-Poincaré operator on planar domains with corners

“…Section 3 generalizes results of [11] to the Lipschitz boundary case. First by using Plemelj's intertwining formula as the main ingredient in the similarity between the adjoint K * of the Neumann-Poincaré operator and two different bounded, selfadjoint operators.…”

“…The arguments essentially follow those of [11], with some additional technicalities, addressed in [9], arising from the fact that K is no longer a compact operator on the scale H s of Besov spaces, 0 ≤ s ≤ 1.…”

“…As expected we have that 0) is the subspace we subtracted from H in order to have a norm. See Section 1 of [11] for proofs which carry over verbatim to our Lipschitz setting for all the statements of this paragraph.…”

“…Following the computations of [11] in our Lipschitz setting we may now realize the angle operator P i (P d − P s )P i as an operator acting on the R n -valued Bergman type space B(Ω), B(Ω) = {∇u ∈ L 2 (Ω) : ∆u = 0}. Defining the operator Π Ω (∇u)(x) = p. v. ∇ x Ω ∇ y G(x, y) · ∇ y u dy, x ∈ Ω, we have the following result.…”

Cauchy, Goursat and Dirichlet Problems for Holomorphic Partial Differential Equations

“…The question of existence of solutions to the Dirichlet problem had an important impact on the development of mathematics in the early 20th century and it is connected with prominent names like H. Poincaré, C. Neumann, D. Hilbert, I. Fredholm and O. Perron, see e.g the exposition [68]. Explicit solutions for the Dirichlet problem can be constructed only for domains of special geometry.…”