Brinkman-type Operators on Riemannian Manifolds: Transmission Problems in Lipschitz and C 1 Domains

Abstract

In this paper we use the method of boundary integral equations to treat some transmission problems for Brinkman-type operators on Lipschitz and C 1 domains in Riemannian manifolds.Keywords Brinkman operator · Stokes operator · Lipschitz and C 1 domains · Riemannian manifold · Transmission problem · Layer potentials · Sobolev spaces Mathematics Subject Classifications (2000) 35J25 · 42B20 · 46E35 · 76D · 76M

Integral representations for solutions of some BVPs for the Lamé system in multiply connected domains

“…Moreover the results obtained for other integral representations for several partial differential equations on domains with lower regularity (see, e.g., the references of [20] for C 1 or Lipschitz boundaries and [21] for "worse" domains) lead one to hope that our approach could be extended to more general domains.…”

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

“…Moreover, in view of the property (see [17]) D + P ;∂Ω − D − P ;∂Ω h 0 ∈ Rν, the first equation in (33) reduces to …”

“…has been introduced in [17] as the pseudodifferential Brinkman operator, while B 0 is the Stokes operator. B P extends to a Fredholm operator with index zero…”

“…Mitrea and Taylor in [29][30][31][32][33] developed the potential theory for elliptic operators on Lipschitz domains in Riemannian manifolds with various smoothness conditions and applied it to the associated boundary value problems (see also [27]). Recently, the authors in [16][17][18][19] considered pseudodifferential Brinkman operators on compact Riemannian manifolds (i.e., variable coefficient operators that extend the differential Brinkman operator from the Euclidean setting to compact Riemannian manifolds), and used the layer potential theory to treat the associated transmission problems on C 1 or, more generally, Lipschitz domains on compact Riemannian manifolds, with the data in L 2 spaces. The study of these transmission problems on the sphere S 2 or other compact surfaces is justified by the flow of water or other viscous incompressible fluids on the Earth, passing through porous rocks or porous soil.…”

Integral equations method and the transmission problem for the stokes system

“…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]). The transmission problem for the Stokes system was studied by the integral equation method in [3], [36].…”