Brezis-Gallouet-Wainger Type Inequality for Irregular Domains

Brezis-Gallouet-Wainger type inequality for irregular domains Vladimir Maz'ya Department of Mathematical Sciences University of Liverpool, M&O Building Liverpool, L69 3BX, UK Department of Mathematics LinköpingUniversity SE-58183 Linköping,Sweden Tatyana Shaposhnikova Department of Mathematics LinköpingUniversity SE-58183 Linköping,Sweden Dedicated to Viktor Burenkov on the occasion of his 70th birthday Abstract. A Brezis-Gallouet-Wainger logarithmic interpolation-embedding inequality is proved for various classes of irregular domains, in particular, for power cusps and λ-John domains. 2010 MSC. Primary: 46E35; Secondary: 32U20 Keywords: logarithmic interpolation inequalities, Sobolev embeddings, capacities 1 Introduction Brezis and Gallouet [BG] established the unique solvability of the initial-boundary value problem for the nonlinear Schrödingerevolution equations with zero Dirichlet data on the boundary of a bounded domain in R2 or its complementary domain. They used crucially the following interpolation-embedding inequality ( ) ( )1=2 kukL1(Ω) ≤ C 1 + log(1 + kukW 2;2(Ω)) (1.1) 2;2 for every u 2 W (Ω) with kukW 1;2(Ω) = 1. Applications of this inequality to the Euler equation can be found in Chapter 13 of M.E. Taylor's book [Ta]. Brezis and Wainger [BW] extended (1.1) to Sobolev spaces of higher order on Rn in the form ( ) ( )(n−k)=n kukL1(Rn) ≤ C 1 + log(1 + kukW l;q (Rn)) (1.2) for every function u in W l;q(Rn) normalized by k k u W k;n=k(Rn) = 1; 1 where k and l are integers, 1 ≤ k < l, ql > n, and k ≤ n. Inequalities (1.1) and (1.2) were studied in different directions in [En], [Go], [Sa]. We prove some inequalities of Brezis-Gallouet-Wainger type for various classes of irregular domains. In particular, classes Jα introduced in [Ma1], power cusps and λ-John domains are considered. 2 Preliminaries n Let Ω be an open domain in R such that mn(Ω) < 1, where mn denotes the n- dimensional Lebesgue measure on Ω. Given p 2 [1; 1) and sets E ⊂ G ⊂ Ω, the capacity Cp(E; G) of the condenser (E; G) is defined as nZ p 1;p Cp(E; G) = inf jruj dx : u 2 L (Ω); u > 1 in E and Ω o u ≤ 0 in ΩnG (up to a set of standard p−capacity zero) : (2.1) Here L1;p(Ω) denotes the Sobolev space defined as { } L1;p(Ω) = u 2 Lp(Ω; loc): u is weakly differentiable in Ω and jruj 2 Lp(Ω) ; endowed with the seminorm kukL1;p(Ω) = krukLp(Ω): We need two isocapacitary functions. The first of them νp : [0; mn(Ω)=2] ! [0; 1] is defined by { ν (s)=inf C (E; G): E and G are measurable subsets of Ω such that p p } E ⊂ G and s ≤ mn(E); mn(G) ≤ mn(Ω)=2 for s 2 [0; mn(Ω)=2]: (2.2) Clearly, the function νp is non-decreasing. The isocapacitary inequality on Ω is a straightforward consequence of definition(2.2) and tells us that νp(mn(E)) ≤ Cp(E; G) (2.3) for every measurable sets E ⊂ G ⊂ Ω with mn(G) < mn(Ω)=2. The second isocapacitary function πp : [0; mn(Ω)=2] ! [0; 1] is given by { π (s)=inf C (E; G): E is a point and G is a measurable subset of Ω p p } such that E 2 G and mn(G) ≤ s for s 2 [0; mn(Ω)=2]: (2.4) The function πp is clearly non-increasing and the corresponding isocapacitary inequality on Ω is πp(mn(G)) ≤ Cp(E; G): (2.5) Variants of these isocapacitary functions were introduced in Chaptrs 3 and 5 of [Ma2] and employed to provide necessary and sufficient conditions for embeddings in the Sobolev space of functions with gradient in Lp. 2 We need another function of purely geometric nature associated with Ω. It is called the isoperimetric function of Ω and will be denoted by λ. The function λ : [0; mn(Ω)=2] ! [0; 1] is given by { } λ(s) = inf P (E): s ≤ mn(E) ≤ mn(Ω)=2 : (2.6) Here, P (E) is the De Giorgi perimeter of E which can be defined as P (E) = Hn−1(@∗E); where @∗E stands for the essential boundary of E in the sense of geometric measure theory and Hn−1 denotes the (n − 1)-dimensional Hausdorff measure. Recall that @∗E agrees with the topological boundary @E of E when E is regular enough, e.g. is an open subset of Ω with a smooth boundary. The very definition of λ leads to the isoperimetric inequality on Ω which reads λ(mn(E)) ≤ P (E) (2.7) for every measurable set E ⊂ Ω with mn(E) ≤ mn(Ω)=2. The isoperimetric function of an open subset of Rn was introduced in [Ma1] (see also [Ma2]) to characterize Sobolev embeddings for functions with gradient in L1. In more recent years, isoperimetric inequalities and corresponding isopertimetric functions have been intensively investigated on Riemannian manifolds as well, see e.g. [BC, CF, CGL, GP, Gr, MHH, Kle, MJ, Pi, Ri]. The functions νp and πp on one hand, and λ on the other hand, are related by the inequalities (Z ) mn(Ω)=2 1−p ≥ dr νp(s) p0 (2.8) s λ(r) and Z ( s )1−p ≥ dr πp(s) p0 (2.9) 0 λ(r) 0 for s 2 (0; mn(Ω)=2) with p = p=(p − 1), which follows along the same lines as in [Ma2], Proposition 4.3.4/1. It is shown in [Ma2], Lemma 3.2.4, that λ(s) > 0 for s > 0. Owing to inequalities (2.8) and (2.9), we have νp(s) > 0 and πp(s) > 0 for s 2 (0; mn(Ω)=2) as well. Clearly, (n−1)=n λ(s) ≤ cn s : (2.10) Moreover, if Ω is bounded and Lipschitz, then λ(s) ≈ s(n−1)=n near s = 0: (2.11) Here and in what follows, the notation f ≈ g near 0 (2.12) for functions f; g : (0; 1) ! [0; 1) means that there exist positive constants c1, c2, and s0 such that c1g(c1s) ≤ f(s) ≤ c2g(c2s) if s 2 (0; s0): (2.13) Furthermore, for any Ω, (n−p)=n νp(s) ≤ cn;p s for n > p (2.14) 3 and ( 1)1−n ν (s) ≤ c log for n = p: (2.15) p n s Both inequalities can be verified by setting appropriate test functions in the above definition of the p-capacity. If Ω is bounded and Lipschitz, then, by (2.11) and (2.8), ( s(n−p)=n if n > p; ≈ ( ) νp(s) 1 1−n (2.16) log s if n = p near s = 0. Similarly to (2.14) and (2.15), we obtain the equality πp(s) = 0 for n ≥ p: (2.17) Finally, if Ω is bounded and Lipschitz, then (n−p)=n πp(s) ≈ s for p > n: (2.18) In what follows we use the notations Gσ = fx 2 Ω: u(x) > σg; Lσ = fx 2 Ω: u(x) < σg; Eσ = fx 2 Ω: u(x) = σg: 3 Main results for arbitrary Ω 1;p 1;r Theorem 1 For every " 2 (0; mn(Ω)=2) and for all u 2 L (Ω) \ L (Ω), p ≥ 1, r ≥ 1, −1=p −1=r oscΩu ≤ πp(") krukLp(Ω) + νp(") krukLr (Ω); (3.1) where oscΩu = ess sup u − ess inf u: Ω Ω Proof. Let the numbers T and t be chosen so that mnfx : u(x) > T g ≤ mn(Ω)=2 ≤ mnfx : u(x) ≥ T g; mnfx : u(x) > tg ≤ " ≤ mnfx : u(x) ≥ tg: Furthermore, let S := ess sup u: Ω Then Z ( ) ( )1=p −1=p p S − t ≤ cp(ES; Gt) jruj dx Gt and Z ( ) ( )1=r −1=r r t − T ≤ cr(Gt; GT ) jruj dx GT nGt which implies −1=p −1=r − ≤ kr k p kr k r ess sup u T πp(") u L (GT ) + νp(") u L (GT ): Ω 4 An analogous estimate for T − ess infΩ u is proved in the same way. Adding both estimates, we arrive at (3.1). We obtain a direct consequence of Theorem 1 and the above lower estimates for the isocapacitary functions which is formulated in terms of the isoperimetric function λ. Corollary 1 Let p ≥ 1, r ≥ 1. Then, for every " 2 (0; mn(Ω)=2) and for all u 2 L1;p(Ω) \ L1;r(Ω) (Z ) 0 (Z ) 0 " dµ 1=p mn(Ω)=2 dµ 1=r ≤ kr k p kr k r oscΩu p0 u L (Ω) + r0 u L (Ω): (3.2) 0 λ(µ) " λ(µ) Proof. The result follows from the lower estimates (5.6) and (5.7) inserted into (3.1). Remark 1. One can add the inequality p > n in Theorem 1 since (3.1) has no sense for n ≥ p because of (2.17). In the case r = 1 the estimate (3.2) is simplified as Z ( " )1=p0 dµ −1 ≤ kr k p kr k 1 oscΩu p0 u L (Ω) + λ(") u L (Ω); (3.3) 0 λ(µ) where p > n. 4 Domains of the class Jα We say that a domain belongs to the class Jα, α > 0, if there is a constant Kα such that α λ(µ) ≥ Kα µ for µ 2 (0; mn(Ω)=2). This class was introduced in [Ma1] and studied in detail in [Ma2], [Ma3]. Corollary 1 can be made more visible for Ω 2 Jα.

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