Poincar\'e inequality on subanalytic sets

Let $\Omega$ be a subanalytic bounded open subset of $\mathbb{R}^n$, with possibly singular boundary. We show that given $p\in [1,\infty)$, there is a constant $C$ such that for any $u\in W^{1,p}(\Omega)$ we have $||u-u_{\Omega}||_{L^p} \le C||\nabla u||_{L^p},$ where we have set $u_{\Omega}:=\frac{1}{|\Omega|}\int_{\Omega} u.$


Introduction
There are several different types of inequalities being called by the name of the great French mathematician. One of them asserts that, given a bounded open subset Ω of R n with Lipschitz boundary and p ∈ [1, ∞), there is a constant C such that for any u ∈ W 1,p (Ω) we have ||u − u Ω || p ≤ C||∇u|| p , where we have set (0.1) u Ω := 1 |Ω| Ω u(x) dx and |Ω| stands for the Lebesgue measure of Ω.
Under this form, it is sometimes called Poincaré-Wirtinger inequality. This inequality plays an important role in the theory of partial differential equations. It is well-known that it is no longer true if we drop the assumption that Ω has Lipschitz boundary. It is actually an interesting problem to study the interplay between the geometry of the singularities of the boundary and this result of analysis [M].
We show in this article that this inequality holds on every subanalytic bounded open subset of R n , with possibly singular boundary. The idea is to use the techniques that the second author recently developed to study L p de Rham cohomology on singular subanalytic varieties [V2] (see Theorem 1.4 below). We do not put any extra ad hoc assumption on the Lipschitz geometry of the boundary.
More generally, the proof could go over any polynomially bounded o-minimal structure [C, D]. It seems that the open sets that are definable in these structures constitute a natural class of singular domains to extend the theory of partial differential equations, and indeed the case of semi-algebraic domains, on which it is possible to carry out effective computations, is already satisfying for most of the applications.
We start by giving definitions and needed facts of subanalytic geometry. Since we do not restrict ourselves to star-shaped domains, the proof of Poincaré inequality will force us to construct homotopies that will not be smooth, which will only be continuous almost everywhere. We thus establish two lemmas that are devoted to the construction of these homotopies, and then carry out the proof of the main theorem.

Subanalytic sets
We now recall some basic facts about subanalytic sets and functions. We refer to [BM] (see also [DS,L,V3]) for proofs and related facts.
Definition 1.1. A subset E ⊂ R n is called semi-analytic if it is locally defined by finitely many real analytic equalities and inequalities. Namely, for each a ∈ R n , there is a neighborhood U of a, and real analytic functions f ij , g ij on U , where i = 1, . . . , r, j = 1, . . . , s i , such that The flaw of the semi-analytic category is that it is not preserved by analytic morphisms, even when they are proper. To overcome this problem, it is convenient to work with a bigger class of sets, the subanalytic sets, which are defined as the projections of the semi-analytic sets: has a neighbourhood U such that U ∩ E is the image under the canonical projection π : R n × R k → R n of some relatively compact semi-analytic subset of R n × R k (where k depends on x). Definition 1.3. We say that a mapping f : A → B is subanalytic, A ⊂ R n , B ⊂ R m , if its graph is a subanalytic subset of R n+m . In the case B = R, we say that f is a subanalytic function.
Subanalytic sets constitute a nice category to study the geometry of semi-analytic sets: it is stable under union, intersection, complement, and Cartesian product. The closure of a subanalytic set is subanalytic. Moreover, these sets enjoy many finiteness properties. For instance, bounded subanalytic sets always have finitely many connected components, each of them being subanalytic.
It is also well-known that they are C 0 triangulable, in the sense that a subanalytic set is always homeomorphic to a simplicial complex. This implies in particular that locally, they have the topology of a cone over what is generally called, the link. This fact is sometimes referred as the local conic structure of the topology of subanalytic sets. Germs of subanalytic sets are nevertheless not bi-Lipschitz homeomorphic to cones, as it is shown by the simple example of a cusp y 2 = x 3 in R 2 . In particular, the cone property which is often used in functional analysis (see [M] for the definition), which is of metric nature, may fail. The theorem below, achieved in [V2], however unravels the Lipschitz properties of the local conic structure. This Lipschitz conic structure was actually derived from techniques developed to construct triangulations that describe the metric properties of singularities [V1] (see also the survey [V3]). Theorem 1.4 (Lipschitz Conic Structure). Let X ⊂ R n be subanalytic and x 0 ∈ X. For ε > 0 small enough, there exists a Lipschitz subanalytic homeomorphism satisfying H |S(x 0 ,ε)∩X = Id, preserving the distance to x 0 , and having the following metric properties: and, for each s ∈ (0, 1], the map r −1 s : B(x 0 , sε) ∩ X → B(x 0 , ε) ∩ X is Lipschitz. Remark 1.5. This theorem is actually valid in every polynomially bounded o-minimal structure, the same proof applying, simply replacing the word "subanalytic" with definable.

Poincaré inequality
For an open subset Ω ⊂ R n and p ≥ 1 we denote by the Sobolev space, where ∂u ∂x i are the partial derivatives of u in the sense of distributions. This space, equipped with the norm is a Banach space. Here, as usual, || · || p stands for the L p norm. It is well known that the set of smooth functions C ∞ (Ω) is dense in W 1,p (Ω) .
We will prove: Theorem 2.1. Let Ω ⊂ R n be a bounded open subanalytic subset. For each p ≥ 1, there exists C > 0 such that for any u ∈ W 1,p (Ω) the following inequality holds where u Ω is as in (0.1).
For the proof, we need two geometric lemmas, the first being necessary to establish the second one. (2) d x h t is invertible f.a.e. (x, t) ∈ Ω × [0, 1], and moreover there exists C > 0 such that whenever d x h t is invertible, we have ||d x h −1 t || ≤ C.
Proof. The proof relies on the following two observations.
(a) In condition (1), possibly composing h with a homothetic transformation, we can choose α as small as we wish. Moreover, as Ω is connected and subanalytic, any two points of Ω can be joint by a subanalytic arc γ. Therefore, possibly composing with a translation of the ball B(z, ε) through an arc γ, the point z can be replaced with any element of Ω. (b) It is enough to construct the desired family of maps on a finite (subanalytic) cover of Ω.
The reason is that, if there exist mappings h : U × [0, 1] → Ω and h ′ : where U, U ′ ⊂ Ω are subanalytic subsets, satisfying (1) and (2) of the lemma, then we can construct a subanalytic mapping h ′′ : (1) and (2) as well. Actually, it is enough to set As the set Ω is compact, by For every s < 1, ν ′ s (u) ≡ 1 − s, and therefore ν s is a bi-Lipschitz homeomorphism with bi-Lipschitz constant L νs which remains bounded if s stays bounded away from 1. Observe also that, since we can argue separately on each connected component of B(x 0 , ε) ∩ Ω (thanks to (b)), it is no loss of generality to assume that this set is connected.
By (b) we may assume that S(x 0 , ε) ∩ Ω is included in a chart of some coordinate system of Ω. By induction on n, we therefore know that there is a family of mappings ,α) for some a ∈ S(x 0 , ε) ∩ Ω andα small enough, such that dh −1 s is bounded uniformly in s. Let us extend trivially (i.e., constantly with respect to the last variable) this family of mappings to a family of mappings, keeping the same notationh We now shall define the desired mapping h by applying successively g andh. Observe for this purpose that r induces a bi-Lipschitz homeomorphisms from S( . Denote by Ψ its inverse and let h s : The inverse of the derivative is bounded by construction. The above lemma makes it possible for us to prove the following lemma which will be useful to establish Theorem 2.1. Let us recall that since subanalytic mappings are differentiable on an open dense subanalytic subset of their domain, they are always differentiable almost everywhere. We use the notation Jac(Γ) to denote the absolute value of the determinant of the Jacobian matrix of a mapping Γ.
Proof. The desired family of arcs will require the following mappings. Let: (1) h : Ω × [0, 1] → Ω be a family of mappings as provided by Lemma 2.2.