REGULARITY OF OPTIMAL SHIP FORMS BASED ON MICHELL'S WAVE RESISTANCE
Résumé
We introduce an optimal shaping problem based on Michell's wave resistance formula in order to find the form of a ship which has an immerged hull with minimal total resistance. The problem is to find a function u ∈ H 1 0 (D), even in the z-variable, and which minimizes the functional J(u) = D |u(x, z)| 2 dxdz + D D k(x, z, x , z)u(x, z)u(x , z)dxdzdx dz with an area constraint on the set {(x, z) ∈ D : u(x, z) = 0} and with the volume constraint D u(x, z)dxdz = V ; D is a bounded open subset of R 2 , symmetric about the x-axis, and k is Michell's kernel. We prove that u is locally α-Hölder continuous on D for all 0 < α < 2/5, and locally Lipschitz continuous on D = {(x, z) ∈ D : z = 0}. The main assumption is the nonnegativity of u. We also prove that the area constraint is " saturated ". The results are first derived for a general kernel k ∈ L q (D × D) with q ∈ (1, +∞].
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