Regularity of Optimal Ship Forms Based on Michell's Wave Resistance

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 is an element of H-0(1) (D), even in the z-variable, and which minimizes the functional J(u) = integral(D)vertical bar del u(x, z)vertical bar(2)dxdz + integral(D)integral(D) k(x, z, x', z')u(x, z)u(x', z')dxdzdx'dz' with an area constraint on the set ((x, z) is an element of D: u(x, z) not equal 0) and with the volume constraint integral(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 alpha-Holder continuous on D for all 0 < alpha < 2/5, and locally Lipschitz continuous on D* = {(x, z) is an element of D : z not equal 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 is an element of L-q (D x D) with q is an element of (1, +infinity]. A numerical simulation illustrates the theoretical result.