تجزیه و تحلیل حساسیت شکلی از مقدار ثابت هاردی

We consider the Hardy constant associated with a domain in the nn-dimensional Euclidean space and we study its variation upon perturbation of the domain. We prove a Fréchet differentiability result and establish a Hadamard-type formula for the corresponding derivatives. We also prove a stability result for the minimizers of the Hardy quotient. Finally, we prove stability estimates in terms of the Lebesgue measure of the symmetric difference of domains.

Let ΩΩ be a bounded domain in RnRn, View the MathML sourcedΩ(x)=dist(x,∂Ω),x∈Ω, and p∈]1,∞[p∈]1,∞[. If there exists c>0c>0 such that equation(1.1) View the MathML source∫Ω|∇u|pdx≥c∫Ω|u|pdΩpdx,for all u∈Cc∞(Ω), Turn MathJax on we then say that the LpLp Hardy inequality holds in ΩΩ. The best constant cc for inequality (1.1) is called the LpLp Hardy constant of ΩΩ and we shall denote it by Hp(Ω)Hp(Ω). It is well-known that if ΩΩ is regular enough then the LpLp Hardy inequality is valid for all p∈]1,∞[p∈]1,∞[; moreover if ΩΩ is convex, and more generally if it is weakly mean convex, i.e. if View the MathML sourceΔdΩ≤0 in the distributional sense in ΩΩ, then Hp(Ω)=((p−1)/p)pHp(Ω)=((p−1)/p)p. The study of inequality (1.1) has a long history which goes back to Hardy himself, see [1]. In the last twenty years there has been a growing interest in the study of Hardy inequalities, the existence and behavior of minimizers [2] and [3], improved inequalities [4] and [5], higher order analogues and other related problems. The precise evaluation of Hp(Ω)Hp(Ω) for domains ΩΩ that are not weakly mean convex is a difficult problem. There are only few examples of such domains for which Hp(Ω)Hp(Ω) is known and these are only for the case p=2p=2 and for very special domains ΩΩ. Even the problem of estimating from below Hp(Ω)Hp(Ω) is difficult and most results again are for p=2p=2. One such result is the well known theorem by A. Ancona which states that H2(Ω)≥1/16H2(Ω)≥1/16 for all simply connected planar domains. We refer to [6], [2], [5], [7], [8], [9] and [10] for more information on the Hardy constant. In this paper we study the variation of Hp(Ω)Hp(Ω) upon variation of the domain ΩΩ. This problem can be considered as a spectral perturbation problem. Indeed, if there exists a minimizer View the MathML sourceu∈W01,p(Ω) for the Hardy quotient associated with (1.1) then uu is a solution to the equation equation(1.2) View the MathML source−Δpu=Hp(Ω)|u|p−2udΩp Turn MathJax on where View the MathML sourceΔpu=div(|∇u|p−2∇u) is the pp-Laplacian. Domain perturbation problems have been extensively studied in the case of the Dirichlet Laplacian as well as for more general elliptic operators, such as operators satisfying other boundary conditions, higher order operators and operators with variable coefficients. We refer to the monographs [11] and [12] for an introduction to this topic. When studying such problems, there are broadly speaking two types of results: qualitative and quantitative. The former provide information such as continuity or analyticity, while the second involve stability properties, possibly together with related estimates. The relevant literature is vast, and we refer to [13], [14], [15], [16] and [17] and references therein for more information; in particular, for the pp-Laplacian we refer to [18], [19] and [20]. In this paper we obtain both qualitative and quantitative results on the domain dependence of Hp(Ω)Hp(Ω). In Theorem 8, we assume that ΩΩ is of class C2C2 with Hp(Ω)<((p−1)/p)pHp(Ω)<((p−1)/p)p and we establish the Fréchet differentiability of Hp(ϕ(Ω))Hp(ϕ(Ω)) with respect to the C2C2 diffeomorphism ϕϕ. In particular we provide a Hadamard-type formula for the Fréchet differential. For our proof we make an essential use of certain results of [2], where it was shown in particular that if Hp(Ω)<((p−1)/p)pHp(Ω)<((p−1)/p)p then the Hardy quotient admits a positive minimizer uu which behaves like View the MathML sourcedΩα near ∂Ω∂Ω for a suitable α>0α>0. In fact, in Theorem 6 we also prove the stability of the minimizer uu in View the MathML sourceW01,p(Ω); this is of independent interest but is also used in the proof of Theorem 8. We subsequently consider stability estimates for Hp(Ω)Hp(Ω). In Theorem 11 we prove under certain assumptions that the Hardy constant Hp(Ω)Hp(Ω) of a C2C2 domain ΩΩ is upper semicontinuous with respect to bi-Lipschitz transformations ϕϕ. In Theorem 12 we consider the stability of the Hardy constant when ΩΩ is subject to a localized perturbation which transforms it to a domain View the MathML sourceΩ̃. Assuming that both ΩΩ and View the MathML sourceΩ̃ are of class C2C2 we obtain stability estimates for the LpLp Hardy constant in terms of the Lebesgue measure of the symmetric difference View the MathML sourceΩ△Ω̃. Estimates of this type have been recently obtained for eigenvalues of various classes of operators; we refer to [21], [14] and [13] and references therein for more information. We finally note that our results are new also for the linear case p=2p=2. The paper is organized as follows. In Section 2 we introduce our notation and prove a general Lipschitz continuity result. Section 3 is devoted to the proof of differentiability results, the Hadamard formula and the stability of minimizers. In Section 4 we prove stability estimates in terms of the Lebesgue measure of the symmetric difference of the domains.