A simple proof of the compactness of the trace operator on a Lipschitz domain
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Archiv der Mathematik
A simple proof of the compactness of the trace operator on a Lipschitz domain ´ment Denis Cle
Abstract. In the setting of bounded strongly Lipschitz domains, we present a short and simple proof of the compactness of the trace operator acting on square integrable vector fields with square integrable divergence and curl with a boundary condition. We rely on earlier trace estimates established in a similar setting. Mathematics Subject Classification. Primary 46E35, Secondary 35B65. Keywords. Lipschitz domains, Trace operators, Divergence theorem.
1. Introduction and main theorem. The compactness of the trace operator on a bounded strongly Lipschitz domain Ω is well known in the scalar case. 1 In fact, it is continuous from H 1 (Ω) to H 2 (∂Ω), and the compactness of the 1 injection H 2 (∂Ω) → L2 (∂Ω) ensures that Tr|∂Ω compactly maps H 1 (Ω) to L2 (∂Ω). By looking coordinate by coordinate, the scalar case implies that the trace operator on H 1 (Ω, Rd ) is also compact. However, for less regular vector fields, the situation is different. In the study of Maxwell and Navier–Stokes equations, it is interesting to study vector fields u ∈ L2 (Ω, Rd ) with div u ∈ L2 (Ω) and curl u ∈ L2 (Ω, Md (R)) with a boundary condition on either the normal part of the trace ν · u or the tangential part of the trace ν ∧ u (for an exact definition of the wedge product, see below at the end of Section 2). A natural boundary condition when considering a fluid is ν · u = 0, and we prove that in this case, the trace operator is still compact. In fact, we were able to prove compactness for a much more general boundary condition where ν · u is only in some Lp spaces with p > 2. We need to introduce a few notations before stating our theorem: let δ > 0 and let
C. Denis
Arch. Math.
XTδ = u ∈ L2 (Ω, Rd ) ; div u ∈ L2 (Ω), curl u ∈ L2 (Ω, M(Rd )), ν · u ∈ L2+δ (∂Ω) and
δ = u ∈ L2 (Ω, Rd ) ; div u ∈ L2 (Ω), curl u ∈ L2 (Ω, M(Rd )) XN ν ∧ u ∈ L2+δ (∂Ω, Md (R)) ,
where Ω is a domain of Rd . δ are Banach spaces when equipped with the natural norms XTδ and XN uXTδ = uL2 (Ω,Rd ) + div uL2 (Ω) + curl uL2 (Ω,Md (R)) + ν · uL2+δ (∂Ω),
(1.1)
uXNδ = uL2 (Ω,Rd ) + div uL2 (Ω) + curl uL2 (Ω,Md (R)) + ν ∧ uL2+δ (∂Ω,Md (R)) .
(1.2)
δ We also denote by XT and XN the subspaces of XTδ and XN with (respectively) ν · u = 0 and ν ∧ u = 0 on ∂Ω. Our aim is to prove the following theorem:
Theorem 1.1. Let δ > 0, let Ω ⊂ Rd be a strongly Lipschitz domain. Then the δ ) to L2 (∂Ω, Rd ). trace operator Tr|∂Ω compactly maps XTδ (or XN An immediate corollary is: Corollary 1.2. The trace operator Tr|∂Ω compactly maps XT and XN into L2 (∂Ω, Rd ). The trace operators on the spaces XT,N are well understood for domains of Rd with regular boundaries. When Ω has a regular enough boundary, the spaces XT and XN are in fact included in H 1 (Ω, Rd ): it was shown for C 1,1 3 domains in dimensions 2 and 3 in [1, Theorems 2.9, 2.12], for C 2 +ε domains (in dimension d) in [6], and for convex domains in [10]. Besides, if Ω is pi
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