The Inhomogeneous Dirichlet Problem for the Stokes System in Lipschitz Domains with Unit Normals Close to VMO

The Inhomogeneous Dirichlet Problem for the Stokes System in Lipschitz Domains with Unit Normals Close to VMO Vladimir Maz’ya, Marius Mitrea and Tatyana Shaposhnikova ∗ Dedicated to the memory of Solomon G. Mikhlin Abstract The goal of this work is to treat the inhomogeneous Dirichlet problem for the Stokes system in a Lipschitz domain Ω ⊆ Rn, n ≥ 2. Our main result is a well-posedness result formulated on the scales of Besov-Triebel-Lizorkin spaces, in the case in which the outward unit normal ν to Ω has small mean oscillation. 1 Introduction The aim of this paper is to discuss the well-posedness of the inhomogeneous Dirichlet problem for the Stokes system of linearized hydrostatics in an arbitrary bounded Lipschitz domain Ω ⊆ Rn, n ≥ 2, when both the solution and the data belong to Besov spaces: ∆~u − ∇π = f~ ∈ Bp,q (Ω), div ~u = g ∈ Bp,q (Ω), s+ 1 −2 s+ 1 −1 p p (1.1) p,q p,q ~ p,q ~u ∈ B 1 (Ω), π ∈ B 1 (Ω), Tr ~u = h ∈ Bs (∂Ω). s+ p s+ p −1 As usual, ~u is the velocity field and π stands for the pressure function. Under the assumption that the outward unit normal ν of Ω has sufficiently small mean oscillation, relative to p, q, s and the Lipschitz constant of ∂Ω, our main result states that the problem (1.1) is uniquely solvable, granted that the data satisfies some necessary compatibility conditions. We are also interested in the case in which the smoothness is measured on the Triebel-Lizorkin scale. More specifically, we have the following result (for notation, definitions and background material see § 2). Theorem 1.1 Let Ω be a bounded Lipschitz domain in Rn, n ≥ 2, of arbitrary topology, and denote by ν, σ, the outward unit normal and surface measure on ∂Ω, respectively. Assume that n−1 1 n < p ≤ ∞, 0 < q ≤ ∞, (n − 1)( p − 1)+ < s < 1, and consider the inhomogeneous Dirichlet problem for the Stokes system with (1.1), subject to the (necessary) compatibility condition Z Z hν,~hi dσ = g(X) dX, for every component O of Ω. (1.2) ∂O O Then there exists δ > 0 which depends only on the Lipschitz character of Ω and the exponent p, with the property that if µ Z Z ¯ ¯ ¶ ¯ ¯ {ν}Osc(∂Ω) := lim sup − − ¯ ν(X) − ν(Y ) ¯ dσ(X)dσ(Y ) < δ, (1.3) ε→0 Bε Bε∩∂Ω Bε∩∂Ω ∗2000 Math Subject Classification. Primary: 35J25, 42B20, 46E35. Secondary 35J05, 45B05, 31B10. Key words: Stokes system, Lipschitz domains, boundary problems, Besov-Triebel-Lizorkin spaces The research of the authors was supported in part by the NSF 1 where the supremum is taken over the collection {Bε} of disks with centers on ∂Ω and of radius ≤ ε, then (1.1) is well-posed (with uniqueness modulo locally constant functions in Ω for the pressure). Hence, in the particular case when the Lipschitz domain Ω is such that ν ∈ VMO (∂Ω), the problem (1.1) is well-posed whenever n−1 1 n < p ≤ ∞, 0 < q ≤ ∞ and (n − 1)( p − 1)+ < s < 1, (1.4) (where (a)+ := max{a, 0}). Consider three Furthermore, the solution has an integral representation formula in terms of hydrostatic layer potential operators and satisfies natural estimates. Concretely, there exists a finite, positive constant C = C(Ω, p, q, s, n) such that k~uk p,q + inf kπ − ck p,q ≤ Ckf~k p,q + Ckgk p,q + Ck~hk p,q , (1.5) B 1 (Ω) c B 1 (Ω) B 1 (Ω) B 1 (Ω) Bs (∂Ω) s+ p s+ p −1 s+ p −2 s+ p −1 where the infimum is taken over all locally constant functions c in Ω. Moreover, analogous well-posedness results hold on the Triebel-Lizorkin scale, i.e. for the problem ~ p,q p,q ∆~u − ∇π = f ∈ F 1 (Ω), div ~u = g ∈ F 1 (Ω), s+ p −2 s+ p −1 p,q p,q p,p (1.6) ~u ∈ F 1 (Ω), π ∈ F 1 (Ω), Tr ~u = ~g ∈ Bs (∂Ω), s+ p s+ p −1 where the data is, once again, made subject to (1.2). This time, in addition to the previous conditions imposed on the indices p, q, it is also assumed that p, q < ∞. p,q Above, the Besov and Triebel-Lizorkin scales As (Ω), A ∈ {B, F }, are defined by restricting n p,q the (tempered) distributions from the corresponding spaces in R to the open set Ω. Also, Bs (∂Ω) stands for the Besov class on the Lipschitz manifold ∂Ω, obtained by transporting (via a partition p,q n−1 of unity and pull-back) the standard scale Bs (R ). The reader should be advised that we make no notational distinction between these smoothness spaces of scalar-valued functions and their natural counterparts for vector-valued functions. It should be noted that conditions (1.4) describe p,q the largest range of indices p, q, s for which the Besov spaces Bs (∂Ω) can be meaningfully defined on the Lipschitz manifold ∂Ω. Regarding the nature of our main result, a few comments are in order. First, no topological restrictions have been imposed on the Lipschitz domain Ω (in particular, the boundary can be disconnected). This is significant since our approach is via boundary layer potentials, whose invertibility properties are directly affected by topological nature of the domain. We overcome this difficulty by devising a suitable combination of such layer potentials. Second, appropriate versions of the above results hold for unbounded Lipschitz domains with compact boundaries, for which (1.3) holds, and for Neumann-type boundary conditions. Third, one can show that actually {ν}Osc(∂Ω) ≈ dist (ν, VMO(∂Ω)), where the distance is taken in BMO(∂Ω). Then condition (1.3) becomes equivalent to dist (ν , VMO (∂Ω)) < δ (1.7) where the distance is measured in BMO (∂Ω). Fourth, codimension one surfaces in Rn whose unit normal has small BMO norm have been studied by S. Semmes in [50] from the perspective of Geometric Measure Theory, whereas, in the context of PDE’s, the class of Lipschitz domains for which the smallness condition (1.3) holds has been first introduced by V. Maz’ya, M. Mitrea and T. Shaposhnikova in [41] (where the authors have established the well-posedness of the inhomogeneous Dirichlet problem for higher-order elliptic systems with bounded, complex-valued, coefficients). Elliptic PDE’s (with Lp boundary data and non-tangential maximal function estimates for the solution) in the class of (two-sided) NTA domains (in the sense of Jerison and Kenig, [25]) with Ahlfors regular boundaries and satisfying the smallness condition (1.7) (with δ depending on p and 2 the natural geometrical characteristics of the domain) have been recently treated by S. Hofmann, M. Mitrea and M. Taylor in [24]. The problem we address in this paper has a long history and the literature dealing with related issues is remarkably rich. When ∂Ω is sufficiently smooth, various cases (typically corresponding to Sobolev spaces with an integer amount of smoothness) have been dealt with by V.A. Solonnikov [54], L. Cattabriga [10], R. Temam [57], Y. Giga [23], W. Varnhorn [58], R. Dautray and J.-L. Lions [18], among others, when ∂Ω is at least of class C2. This has been subsequently extended by C. Amrouche and V. Girault [6] to the case when ∂Ω ∈ C1,1 and, further, by G.P. Galdi, C.G. Simader and H. Sohr [21] when ∂Ω is Lipschitz, with a small Lipschitz constant. There is also a wealth of results related to Theorem 1.1 in the case when Ω is a polygonal domain in R2, or a polyhedral domain in R3. An extended account of this field of research can be found in V.A. Kozlov, V.G. Maz’ya and J. Rossmann’s monograph [35], which also contains pertinent references to earlier work. Here we also wish to mention the recent work by V. Maz’ya and J. Rossmann [42]. In the case of a bounded Lipschitz domain Ω ⊆ Rn, n ≥ 2, the Dirichlet and Regularity problem, with Lp nontangential maximal function estimates for the solution, have been solved when |p − 2| is small by E.B. Fabes, C.E. Kenig and G.C. Verchota in [20], and when n = 3 and 2 − ε < p < ∞ by Z. Shen in [52]. Higher dimensional versions of the results in [52] have been worked out by J. Kilty in [31]. More recently, the homogeneous and inhomogeneous problem for the Stokes system equipped with Dirichlet, Neumann and transmission boundary conditions in arbitrary Lipschitz domains in Rn, n ≥ 2, has been solved by M. Mitrea and M. Wright in [47]. To describe the portion of this work pertaining to the case of the inhomogeneous Dirichlet problem, for each ε ∈ (0, 1] and n ≥ 2 consider the two-dimensional region Rn,ε in the (s, 1/p)-plane, which depends on the dimension as follows: 1 p q 3 (1, 2 + ε) 1 1 666 ¡¡ p p 1 ¡ 1 ( 2 -ε, 1) slope 666 q 1 666 slope 1 ¡ 2 (1−2ε,1)"" (1,1+ε) slope n−1 1 q QQ 1 QQ ¡ q" Q 1 Q " Q n+1 Qs¡ Q " (1, +ε) 1 1 s Qs ((q 2(n−1) + ε +ε " 1 ((( 2 q¡ 2 "q 2 +ε (q ( 1 1 1 1 2 q (1, -ε) 2 q (1, 1 − ε) 2 (1, 1 − ε) ¡ 2 " 2 ((q 2 ¡Y " n−3 ((( HH slope 1 " 2(n−1) -ε (q ( q¡ -- " -- YH 1 q YHH - H -- O +ε 1 s O 2ε H 1 s O H 1 s 2 H H slope 1 slope 1 Figure 1: Figure 2: 2 Figure 3: n−1 Rn,ε for n = 2 Rn,ε for n = 3 Rn,ε for n ≥ 4 It has been shown in [47] that, given a bounded Lipschitz domain Ω ⊆ Rn, n ≥ 2, there exists ε = ε(Ω) ∈ (0, 1] such that the problem (1.1) (subject to the compatibility conditions (1.2)) is well-posed whenever 0 < q ≤ ∞ and the pair (s, 1/p) belongs to the region Rn,ε, depicted above.

Load more