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

Abstract

We prove the well-posedness of the transmission problem for the Laplacian across a Lipschitz interface, with optimal non-tangential maximal function estimates, for data in Lebesgue and Hardy spaces on the boundary. As a corollary, we show that the spectral radius of the (adjoint) harmonic double layer potential K Ã in L p 0 ð@OÞ is less than 1 2 ; whenever O is a bounded convex domain and 1opp2: r

Dirichlet‐transmission problems for pseudodifferential Brinkman operators on Sobolev and Besov spaces associated to Lipschitz domains in Riemannian manifolds

“…• The complementary hypersingular layer potential operator [8]). Next, we show only the compactness of the complementary single-layer operator V P,0;∂Ω given by (20), since the compactness of the other complementary layer operators follows with similar arguments.…”

“…Mitrea and Wright [34] used layer potential methods to study the main boundary value problems for the Stokes system in Lipschitz domains in R n , n ≥ 2, and in Sobolev and Besov spaces (see also [11] in the case of NTA domains, in the sense of Jerison and Kenig [13], with Ahlfors regular boundaries and small mean oscillations of the unit normals). Escauriaza and Mitrea [8] studied the transmission problem for the Laplace equation in complementary Lipschitz domains in R n , n ≥ 2, with optimal non-tangential maximal function estimates and the data in Lebesgue and Hardy spaces. Various layer potential methods have been also used in [20,22,28] to study boundary value problems for the Stokes or Brinkman equations in Lipschitz domains in Euclidean setting.…”

Counterexamples to the well-posedness of 𝐿^{𝑝} transmission boundary value problems for the Laplacian

“…A domain Ω ⊂ R 2 lying above the graph of a Lipschitz function φ : R → R is called a special Lipschitz domain. That is, (2) Ω :…”

“…Furthermore, ∂ ν stands for the normal derivative on ∂Ω. Recently, in [2], the authors show that, for any γ ∈ (0, 1), the problem (T BV P ) is well posed (uniqueness understood modulo constants) in the class of (special) Lipschitz domains for every p ∈ (1,2]. In this note we answer the question, posed to us by L. Escauriaza and M. Mitrea, of whether this range is optimal in the class of domains under consideration.…”

Integral equations method and the transmission problem for the stokes system

“…This problem occurs in the case of contact of two media with different material constants. Very fruitful is to study this problem using the integral equation method (see [4], [35], [12], [11], [13], [34], [5]). Many papers study the Brinkman transmission problem and the Stokes-Brinkman transmission problem by the integral equation method ( [21], [22], [23], [25], [20], [24]).…”