We give a proof of the Poincaré inequality in W 1,p (Ω) with a constant that is independent of Ω ∈ U, where U is a set of uniformly bounded and uniformly Lipschitz domains in R n . As a byproduct, we obtain the following : The first non vanishing eigenvalues λ 2 (Ω) of the standard Neumann (variational) boundary value problem on Ω for the Laplace operator are bounded below by a positive constant if the domains Ω vary and remain uniformly bounded and uniformly Lipschitz regular.
To cite this version:A Boulkhemair. On a shape derivative formula in the Brunn-Minkowski theory. 2015. hal-01140162On a shape derivative formula in the Brunn-Minkowski theory A. Boulkhemair Laboratoire de Mathématiques Jean Leray, CNRS UMR6629/ Université de Nantes, 2, rue de la Houssinière, BP 92208, 44322 Nantes, France.e-mail : boulkhemair-a arobase univ-nantes.fr
AbstractWe extend a formula for the computation of the shape derivative of an integral cost functional with respect to a class of convex domains, using the so called support functions and gauge functions to express it. This is a priori a formula in shape optimization theory. However, the result also happens to be an extension of a well known formula from the Brunn-Minkowski theory of convex bodies.
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