The proof of this normalized energy decay follows the outline of [7], which is an argument by contradiction. One starts with a sequence u, of p-minimizers on B, which do not exhibit energy decay but which have total p-energies E{ = /@ Iv u,IP dx approaching 0 as i + 00.

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We complete the details of a theory outlined by Kontsevich and Soibelman that associates to a semi-algebraic set a certain graded commutative differential algebra of "semi-algebraic differential forms" in a functorial way. This algebra encodes the real homotopy type of the semi-algebraic set in the spirit of the DeRham algebra of differential forms on a smooth manifold. Its development is needed for Kontsevich's proof of the formality of the little cubes operad. P A 6.1. Poincaré Lemma for Ω * P A 6.2. Sheaf propery of Ω * P A 6.3. Extendability of the simplicial set Ω P A (∆ • ) 6.4. The weak equivalence Ω * P A ≃ A P L 7. Monoidal equivalences 8. Oriented semi-algebraic bundles and integration along the fiber 8.1. Properties of SA bundles 8.2. Properties of integration along the fiber