Asymptotics of solutions of the Neumann problem in a domain with closely posed components of the boundary

Abstract

The Neumann problem for the Poisson equation is considered in a domain Ωε ⊂ R n with boundary components posed at a small distance ε > 0 so that in the limit, as ε → 0 + , the components touch each other at the point O with the tangency exponent 2m 2. Asymptotics of the solution uε and the Dirichlet integral ∇xuε; L 2 (Ωε) 2 are evaluated and it is shown that main asymptotic term of uε and the existence of the finite limit of the integral depend on the relation between the spatial dimension n and the exponent 2m. For example, in the case n < 2m − 1 the main asymptotic term becomes of the boundary layer type and the Dirichlet integral has no finite limit. Some generalizations are discussed and certain unsolved problems are formulated, in particular, non-integer exponents 2m and tangency of the boundary components along smooth curves. Therefore, with the error defined by the majorant in estimate (3.49), not smaller than all remaining errors, the scalar product J 3 in (3.40) can be replaced by the scalar product (3.43), which by obvious reasons (in particular, owing to χχ 0 = χ 0 ) coincides with(3.52)We return to the analysis of the first three terms on the right-hand side of (3.36), we have already rewritten two of them as follows:J 7 = (f , ψ) Ωε + (g, ψ) Γε ,

Justification of lubrication approximation: An application to fluid/solid interactions

“…We now wish to apply Proposition 11 to q in in order to yield a constant that bounds its H 3 -norm by something which does not depend on h. However, one term in the right-hand side of (69) depends on h. To avoid this difficulty we expand q in = q (1) in + hq (2) in with pressure field components defined as respective solutions to:…”

“…In particular, we remark that q (1) in is a solution to (26)-(27) with a source term f given by (28) associated with w * ⊥ = u * ⊥ + γ b divu * // v * = u * // , and that q (2) in is a solution to (26)-(27) with a source term f given by (28) associated with…”

“…So, we split again q in = q (1) in + hq (2) in with q (1) in and q (2) in defined respectively as the solutions to…”

“…with K depending on C reg 3 , C cvx , C ell and L. As for q (1) in , we remark that it is a solution to (26)-(27) with a source term f given by (28) associated with w * ⊥ = u * R,⊥ + γ b div x,y u * R,// v * = u * R,// . In this case, we have that w * ⊥ (0) = 0 and also that ∇w * ⊥ (0), v * (0) vanish.…”

Well‐posedness of the Laplacian on manifolds with boundary and bounded geometry

“…It was shown in [52] that every Riemannian manifold is conformal to a manifold with bounded geometry. Moreover, manifolds of bounded geometry can be used to study boundary value problems on singular domains, see, for instance, [17,18,22,23,38,46,[53][54][55] and the references therein.…”