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

  • Julien Dambrine
  • Morgan Pierre


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 \(u\in H^1_0(D)\), even in the z-variable, and which minimizes the functional
$$\begin{aligned} J(u)=\int _D|\nabla u(x,z)|^2dxdz+\int _D\int _Dk(x,z,x',z')u(x,z)u(x',z')dxdzdx'dz' \end{aligned}$$
with an area constraint on the set \(\{(x,z)\in D\ :\ u(x,z)\not =0\}\) and with the volume constraint \(\int _D u(x,z)dxdz=V\); D is a bounded open subset of \(\mathbb {R}^2\), symmetric about the x-axis, and k is Michell’s kernel. We prove that u is locally \(\alpha \)-Hölder continuous on D for all \(0<\alpha <2/5\), and locally Lipschitz continuous on \(D^\star =\{(x,z)\in D\ : z\not =0\}\). 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 \(k\in L^q(D\times D)\) with \(q\in (1,+\infty ]\). A numerical simulation illustrates the theoretical result.


Shape optimization Existence Regularity Dirichlet energy Wave resistance 

Mathematics Subject Classification

49N60 76B75 49J40 49Q10 



The authors have been partially supported by the “Action Concertée Incitative: Opti-Ondes (2015–2016)” of the University of Poitiers. The authors also acknowledge the group “Phydromat”, and Germain Rousseaux in particular, for stimulating discussions. The second author is thankful to Michel Pierre for helpful discussions.


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Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et Applications, UMR CNRS 7348Université de PoitiersFuturoscope ChasseneuilFrance

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