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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 \(u\in H^1_0(D)\), 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.

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  1. 1.

    Aguilera, N., Alt, H.W., Caffarelli, L.A.: An optimization problem with volume constraint. SIAM J. Control Optim. 24(2), 191–198 (1986)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Alcevedo, M.L., Mazarredo, L. (eds.): Proceedings of the 8th International Towing Tank Conference, Madrid, Spain, El Pardo, Madrid (1957)

  3. 3.

    Allaire, G.: Conception optimale de structures. Mathématiques & Applications, vol. 58. Springer, Berlin (2007)

    MATH  Google Scholar 

  4. 4.

    Alt, H.W., Caffarelli, L.A.: Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325, 105–144 (1981)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Bony, J.-M.: Cours d’analyse—Théorie des distributions et analyse de Fourier. Ecole Polytechnique, Palaiseau (Essonne) (2001)

    MATH  Google Scholar 

  6. 6.

    Bourbaki, N.: Éléments de mathématique. Fasc. XXXIII. Variétés différentielles et analytiques. Fascicule de résultats (Paragraphes 1 à 7). Actualités Scientifiques et Industrielles, No. 1333. Hermann, Paris (1967)

  7. 7.

    Brezis, H.: Analyse fonctionnelle. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris (1983)

    Google Scholar 

  8. 8.

    Briancon, T.: Regularity of optimal shapes for the Dirichlet’s energy with volume constraint. ESAIM Control Optim. Calc. Var. 10(1), 99–122 (2004)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Briançon, T., Hayouni, M., Pierre, Mi: Lipschitz continuity of state functions in some optimal shaping. Calc. Var. Partial Differ. Equ. 23(1), 13–32 (2005)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations and Their Applications, vol. 65. Birkhäuser Boston Inc, Boston, MA (2005)

    Book  Google Scholar 

  11. 11.

    Crouzeix, M.: Variational approach of a magnetic shaping problem. Eur. J. Mech. B Fluids 10(5), 527–536 (1991)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Dambrine, J., Pierre, Mo., Rousseaux, G.: Shape optimization of ship hulls based on Michell’s and Sretensky’s formulas (in preparation)

  13. 13.

    Dambrine, J., Pierre, Mo., Rousseaux, G.: Optimization of ship hulls considered as slender bodies. In: Hydrodynamics applied to inland waterways and port approaches, Paris-Meudon. SHF and AIPCN (November 2015)

  14. 14.

    Dambrine, J., Pierre, Mo, Rousseaux, G.: A theoretical and numerical determination of optimal ship forms based on Michell’s wave resistance. ESAIM Control Optim. Calc. Var. 22(1), 88–111 (2016)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Textbooks in Mathematics, revised edn. CRC Press, Boca Raton, FL (2015)

    Book  Google Scholar 

  16. 16.

    Giaquinta, M., Modica, G., Souček, J.: Cartesian currents in the calculus of variations. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 37. Springer, Berlin (1998)

  17. 17.

    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin (2001)

    Book  Google Scholar 

  18. 18.

    Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhäuser Verlag, Basel (1984)

    Book  Google Scholar 

  19. 19.

    Gotman, ASh: Study of Michell’s integral and influence of viscosity and ship hull form on wave resistance. Ocean. Eng. Int. 6, 74–115 (2002)

    Google Scholar 

  20. 20.

    Gustafsson, B., Shahgholian, H.: Existence and geometric properties of solutions of a free boundary problem in potential theory. J. Reine Angew. Math. 473, 137–179 (1996)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Havelock, T.H.: The theory of wave resistance. Proc. R. Soc. Lond. A 132(835) (1932)

  22. 22.

    Hecht, F.: New development in freefem++. J. Numer. Math. 20(3–4), 251–265 (2012)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Henrot, A.: Shape Optimization and Spectral Theory. De Gruyter Open, Berlin (2017)

    Book  Google Scholar 

  24. 24.

    Henrot, A., Pierre, Mi: Variation et Optimisation de Formes. Mathématiques & Applications, vol. 48. Springer, Berlin (2005)

    Book  Google Scholar 

  25. 25.

    Hsiung, C.-C.: Optimal ship forms for minimum wave resistance. J. Ship Res. 25(2), 95 (1981)

    Google Scholar 

  26. 26.

    John, F.: Plane Waves and Spherical Means Applied to Partial Differential Equations. Interscience Publishers, New York (1955)

    MATH  Google Scholar 

  27. 27.

    Kinderlehrer, D., Nirenberg, L.: Regularity in free boundary problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4(2), 373–391 (1977)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Kostyukov, A.A.: Theory of Ship Waves and Wave Resistance. Effective Communications Inc., Iowa City (1968)

    Google Scholar 

  29. 29.

    Krein, M.G., Sizov, V.G.: On the form of a ship of minimum total resistance (in Russian) unpublished (1960)

  30. 30.

    Landais, N.: A regularity result in a shape optimization problem with perimeter. J. Convex Anal. 14(4), 785–806 (2007)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Lian-en, Z.: Optimal ship forms for minimal total resistance in shallow water. Schriftenreihe Schiffbau 445, 1–60 (1984)

    Google Scholar 

  32. 32.

    Michell, J.H.: The wave resistance of a ship. Philos. Mag. 5(45), 106–123 (1898)

    Article  Google Scholar 

  33. 33.

    Michelsen, F.C.: Wave resistance solution of Michell’s integral for polynomial ship forms. PhD thesis, University of Michigan, (1960)

  34. 34.

    Sizov, V.G.: The seminar on ship hydrodynamics, organized by Professor M. G. Krein. In: Differential Operators and Related Topics, Vol. I (Odessa, 1997), Oper. Theory Adv. Appl., vol. 117, pp. 9–20. Birkhäuser, Basel (2000)

  35. 35.

    Sretensky, L.N.: On the wave-making resistance of a ship moving along in a canal. Philos. Mag. 22, 1005–1013 (1936)

    Article  Google Scholar 

  36. 36.

    Sretensky, L .N.: Sur la détermination de la résistance ondulatoire d’un navire se déplaçant à la surface de l’eau d’une profondeur finie. C. R. Acad. Sci. l’URSS 2((11)(90)), 265–267 (1936)

    MATH  Google Scholar 

  37. 37.

    Tuck, E., Lazauskas, L.: Drag on a ship and Michell’s integral. In: Proceedings of the XXII International Congress of Theoretical and Applied Mechanics, Adelaide, Australia (2008)

  38. 38.

    Tuck, E.O.: The wave resistance formula of J. H. Michell (1898) and its significance to recent research in ship hydrodynamics. J. Austral. Math. Soc. Ser. B 30(4), 365–377 (1989)

    MathSciNet  Article  Google Scholar 

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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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Correspondence to Morgan Pierre.


Michell’s Wave Resistance Kernel

From (7.1) and (7.3), by (formally) inverting the integrals, we see that Michell’s normalized wave resistance can be written

$$\begin{aligned} J_{wave}(u)=\int _{D\times D}k_\nu (x,z,x',z')u(x,z)u(x',z')dxdzdx'dz' \end{aligned}$$


$$\begin{aligned} k_\nu (x,z,x',z')=\frac{4\nu ^4}{\pi C_F(\nu )}K(\nu (x-x'),\nu (|z|+|z'|)), \end{aligned}$$


$$\begin{aligned} K(X,Z)=\int _1^\infty e^{-\lambda ^2Z}\cos (\lambda X)\frac{\lambda ^4}{\sqrt{\lambda ^2-1}}d\lambda . \end{aligned}$$

This formal computation will be made rigorous below (see Corollary A.2). This expression of Michell’s resistance in terms of a kernel \(k_\nu \) is well-known [28], but to the best of our knowledge, the results in Appendix A are new.

First notice that K is defined and continuous on \(\mathbb {R}\times (0,+\infty )\) and

$$\begin{aligned} |K(X,Z)|\le I(Z)<+\infty \end{aligned}$$

for all \((X,Z)\in \mathbb {R}\times (0,+\infty )\), with

$$\begin{aligned} I(Z)=\int _1^\infty e^{-Z\lambda ^2}\frac{\lambda ^4}{\sqrt{\lambda ^2-1}}d\lambda . \end{aligned}$$

In particular, \(k_\nu \) is continuous on \((\mathbb {R}\times \mathbb {R}^\star )^2\).

The following result is essential for the Hölder regularity of the optimal ship.

Theorem A.1

Michell’s normalized wave resistance kernel \(k_\nu \) (A.2) belongs to \(L^q(D\times D)\) for all \(1\le q<5/4\). Moreover, if D contains an open disc centered on the x-axis, then \(k_\nu \) does not belong to \(L^{5/4}(D\times D)\).


It is sufficient to prove the assertion for a domain D of the form \(D_l=(-l,l)\times (-l,l)\) where \(l>0\) is arbitrary. Moreover, by the change of variable \((x,z,x',z')\rightarrow (\nu x,\nu z,\nu x',\nu z')\) in (A.1)–(A.2), it will suffice to consider the case \(\nu =1\). We write

$$\begin{aligned} K(X,Z)=I_1(X,Z)+I_2(X,Z)+I_3(X,Z)+I_4(X,Z), \end{aligned}$$


$$\begin{aligned} I_1(X,Z)= & {} \int _1^2e^{-\lambda ^2 Z}\cos (\lambda X)\frac{\lambda ^4}{\sqrt{\lambda ^2-1}}d\lambda ,\\ I_2(X,Z)= & {} \int _2^\infty e^{-\lambda ^2Z}\cos (\lambda X)\left( \frac{\lambda ^4}{\sqrt{\lambda ^2-1}}-\lambda ^3\right) d\lambda ,\\ I_3(X,Z)= & {} -\int _0^2e^{-\lambda ^2Z}\cos (\lambda X)\lambda ^3d\lambda ,\\ I_4(X,Z)= & {} \int _0^\infty e^{-\lambda ^2Z}\cos (\lambda X)\lambda ^3d\lambda . \end{aligned}$$

By Lebesgue’s dominated convergence theorem, \(I_1\) and \(I_3\) are continuous on \(\mathbb {R}^2\). We will prove that \((x,z,x',z')\mapsto I_2(x-x',|z|+|z'|)\) belongs to \(L^{5/4}(D_l\times D_l)\) and that \((x,z,x',z')\mapsto I_4(x-x',|z|+|z'|)\) belongs to \(L^q(D_l\times D_l)\) for all \(1\le q<5/4\), but does not belong to \(L^{5/4}(D_l\times D_l)\). The theorem will then be proved.

By the mean value theorem, there exists \(C_1>0\) such that

$$\begin{aligned} 0\le \frac{1}{\sqrt{1-u}}-1\le C_1u,\quad \forall u\in [0,1/4]. \end{aligned}$$

Thus, for all \(\lambda \ge 2\),

$$\begin{aligned} 0\le \frac{1}{\sqrt{1-1/\lambda ^2}}-1\le \frac{C_1}{\lambda ^2}, \end{aligned}$$

and so

$$\begin{aligned} 0\le \frac{\lambda ^4}{\sqrt{\lambda ^2-1}}-\lambda ^3\le C_1\lambda . \end{aligned}$$

We obtain

$$\begin{aligned} |I_2(X,Z)|\le C_1\int _0^\infty e^{-\lambda ^2Z}\lambda d\lambda . \end{aligned}$$

Performing the change of variable \(\mu =\sqrt{Z}\lambda \), we find

$$\begin{aligned} |I_2(X,Z)|\le \frac{C_1}{Z}\int _0^\infty e^{-\mu ^2}\mu d\mu \le \frac{C_1'}{Z}. \end{aligned}$$

Next, we notice that for \(q>1\), the integral \(\int _0^l\int _0^l(z+z')^{-q}dzdz'\) is finite if and only if \(q<2\). Indeed,

$$\begin{aligned} \int _0^l\int _0^l(z+z')^{-q}dzdz= & {} \frac{1}{q-1}\int _0^l[z^{1-q}-(z+l)^{1-q}]dz \end{aligned}$$
$$\begin{aligned}\le & {} \frac{l^{2-q}}{(q-1)(2-q)}<\infty \end{aligned}$$

if \(q<2\), whereas the integral on the right-hand side of (A.8) is \(+\infty \) if \(q\ge 2\). In particular, for \(q=5/4\), the function \((x,z,x',z')\mapsto 1/(|z|+|z'|)\) belongs to \(L^{5/4}(D_l\times D_l)\) since

$$\begin{aligned} \int _{D_l\times D_l}\frac{1}{(|z|+|z'|)^q}dxdzdx'dz'=16l^2\int _0^l\int _0^l\frac{1}{(z+z')^q}dzdz'<\infty . \end{aligned}$$

By (A.7), the function \((x,z,x',z')\mapsto I_2(x-x',|z|+|z'|)\) belongs to \(L^{5/4}(D_l\times D_l)\) as well.

Concerning the term \(I_4\), we first perform the change of variable \(\mu =\sqrt{Z}\lambda \), so that

$$\begin{aligned} I_4(X,Z)=\frac{1}{Z^2}J\left( \frac{X}{\sqrt{Z}}\right) , \end{aligned}$$

with \(J(t)=\int _0^\infty e^{-\mu ^2}\cos (t\mu )\mu ^3d\mu \). By Lebesgue’s dominated convergence theorem, J is continuous on \(\mathbb {R}\); in particular, J is bounded on \([-1,1]\) by a constant \(C_2\). Integration by parts yields

$$\begin{aligned} J(t)=-\frac{1}{t}\int _0^\infty \sin (t\mu )\left( 3\mu ^2-2\mu ^4\right) e^{-\mu ^2}d\mu , \end{aligned}$$

so that

$$\begin{aligned} |J(t)|\le \frac{1}{|t|}\int _0^\infty \left( 3\mu ^2+2\mu ^4\right) e^{-\mu ^2}d\mu =\frac{C_3}{|t|}, \end{aligned}$$

for all \(t\not =0\). Let now \(q>1\). On performing the linear change of variable \(X=x-x'\), \(X'=x+x'\), we find that

$$\begin{aligned}&\int _{D_l\times D_l}|I_4(x-x',|z|+|z'|)|^qdxdzdx'dz'\\&\quad \le 2l\int _{-l}^l\int _{-l}^l\int _{-2l}^{2l}|I_4(X,|z|+|z'|)|^qdXdzdz'\\&\quad =8l\int _{0}^l\int _{0}^l\int _{-2l}^{2l}|I_4(X,z+z')|^qdXdzdz'\\&\quad =8l\int _{0}^l\int _{0}^l\int _{-2l}^{2l}\frac{1}{(z+z')^{2q}}\left| J\left( \frac{X}{\sqrt{z+z'}}\right) \right| ^qdXdzdz'. \end{aligned}$$

Integration with respect to X yields

$$\begin{aligned} \int _{-2l}^{2l}\left| J\left( \frac{X}{\sqrt{z+z'}}\right) \right| dX\le & {} \int _{|X|\le \sqrt{z+z'}}\left| J\left( \frac{X}{\sqrt{z+z'}}\right) \right| dX\\&+\int _{ \sqrt{z+z'}\le |X|\le 2l}\left| J\left( \frac{X}{\sqrt{z+z'}}\right) \right| dX\\\le & {} \int _{|X|\le \sqrt{z+z'}}C_2^qdX+2\int _{\sqrt{z+z'}\le X\le 2l}C_3^q\frac{\sqrt{z+z'}^q}{X^q}dX\\\le & {} 2C_2^q\sqrt{z+z'}+\frac{2C_3^q}{q-1}\sqrt{z+z'}\\\le & {} C_4\sqrt{z+z'}. \end{aligned}$$


$$\begin{aligned} \int _{D_l\times D_l}|I_4(x-x',|z|+|z'|)|^qdxdzdx'dz'\le 8lC_4\int _{0}^l\int _0^l(z+z')^{1/2-2q}dzdz'. \end{aligned}$$

The right-hand side is finite if and only if \(q<5/4\) (see (A.8)–(A.9)). This shows that \(I_4\) belongs to \(L^q(D_l\times D_l)\) for all \(1\le q<5/4\), as claimed.

To see the optimality of this statement, first note that \(J(0)>0\) and let \(t_0>0\) such that \(J(t)\ge J(0)/2\) for all \(t\in [-t_0,t_0]\). The linear change of variable \(X=x-x'\), \(X'=x+x'\) maps the square \((-l,l)\times (-l,l)\) onto a square with vertices (2l, 0), (0, 2l), \((-2l,0)\) and \((0,-2l)\), which contains the square \((-l,l)\times (-l,l)\). Let \(q=5/4\). We have

$$\begin{aligned}&\int _{D_l\times D_l}|I_4(x-x',|z|+|z'|)|^qdxdzdx'dz'\\&\quad \ge 4l\int _0^l\int _0^l\frac{1}{(z+z')^{2q}}\left| J\left( \frac{X}{\sqrt{z+z'}}\right) \right| ^qdXdzdz'. \end{aligned}$$

By choosing \(t_0>0\) small enough so that \(t_0\sqrt{2l}\le l\), we also have

$$\begin{aligned} \int _{|X|\le l}\left| J\left( \frac{X}{\sqrt{z+z'}}\right) \right| ^qdX\ge & {} \int _{|X|\le t_0\sqrt{z+z'}}\left| J\left( \frac{X}{\sqrt{z+z'}}\right) \right| ^qdX\\\ge & {} \left( \frac{J(0)}{2}\right) ^q(2t_0\sqrt{z+z'}). \end{aligned}$$

We obtain

$$\begin{aligned} \int _{D_l\times D_l}|I_4(x-x',|z|+|z'|)|^qdxdzdx'dz'\ge 8lt_0\left( \frac{J(0)}{2}\right) ^q\int _0^l\int _0^l(z+z')^{1/2-2q}dzdz'. \end{aligned}$$

The integral on the right-hand side is \(+\infty \) for \(q=5/4\) (see (A.8)). This concludes the proof. \(\square \)

As a consequence, we have:

Corollary A.2

For every \(q'>5\) and for all \(u\in L^{q'}(D)\), the formulations for \(J_{wave}(u)\) given by (7.1)–(7.2) and (A.1)–(A.3) are equal (and finite).


Without loss of generality, we may assume that \(\nu =1\) and \(D=D_l=(-l,l)\times (-l,l)\) with \(l>0\). Let \(q'>5\), \(u\in L^{q'}(D_l)\) and \(q\in (1,5/4)\) such that \(1/q+1/q'=1\). We use the form \(S_u(\lambda )=i\lambda T_u(\lambda )\) with \(T_u(\lambda )=\int _Du(x,z)e^{-i\lambda x}e^{-\lambda ^2 |z|}dxdz\) (cf. (7.3)). For any integer \(N\ge 2\), we have

$$\begin{aligned} \int _1^N|T_u(\lambda )|^2\frac{\lambda ^4}{\sqrt{\lambda ^2-1}}d\lambda =\int _{D_l\times D_l}K_N(x-x',|z|+|z'|)u(x,z)u(x',z')dxdzdx'dz', \end{aligned}$$


$$\begin{aligned} K_N(X,Z)=\int _1^Ne^{-\lambda ^2Z}\cos (\lambda (x-x'))\frac{\lambda ^4}{\sqrt{\lambda ^2-1}}d\lambda . \end{aligned}$$

This is obtained by applying Fubini’s theorem for the two variables \(\lambda \in (1,N)\) and \((x,z,x',z')\in D_l\times D_l\). By the monotone convergence theorem, the left-hand side of (A.11) tends to

$$\begin{aligned} \int _1^\infty |T_u(\lambda )|^2\frac{\lambda ^4}{\sqrt{\lambda ^2-1}}d\lambda \end{aligned}$$

when N tends to \(+\infty \). The convergence of the right-hand side is more delicate. We will apply Lebesgue’s dominated convergence theorem. We first note that \(K_N(x-x',|z|+|z'|)\) tends everywhere in \((D_l\times D_l)\cap (\mathbb {R}\times \mathbb {R}^\star )^2\) to \(K(x-x',|z|+|z'|)\) (see (A.3)). By arguing as in the proof of Theorem A.1, we can show that

$$\begin{aligned} |K_N(x-x',|z|+|z'|)|\le k^\star (x,z,x',z') \text{ in } D_l\times D_l, \end{aligned}$$

where \(k^\star \in L^q(D_l\times D_l)\) is independent of N (details are left to the reader). This implies that the right-hand side of (A.11) tends to

$$\begin{aligned} \int _{D_l\times D_l}K(x-x',|z|+|z'|)u(x,z)u(x',z')dxdzdx'dz' \end{aligned}$$

as N tends to \(+\infty \), by dominated convergence. \(\square \)

Proposition A.3

Assume that \(D\subset \{(x,z)\in \mathbb {R}^2\ :\ |z|>\delta \}\) for some \(\delta >0\). Then \(k_\nu \) (cf. (A.2)) belongs to \( L^\infty (D\times D)\) and

$$\begin{aligned} \Vert k_\nu \Vert _{L^\infty (D\times D)}\le \frac{4\nu ^4}{\pi C_F(\nu )}e^{-\nu \delta }I(\nu \delta ), \end{aligned}$$

where \(I:(0,+\infty )\rightarrow (0,+\infty )\) is the continuous and decreasing function defined by (A.5). In particular, if \(C_F(\nu )^{-1}\) has at most a polynomial growth as \(\nu \) tends to \(+\infty \), then \(\Vert k_\nu \Vert _{L^\infty (D\times D)}\rightarrow 0\) as \(\nu \rightarrow +\infty \).


Let \((x,z,x',z')\in D\times D\), and define \(X=\nu (x-x')\), \(Z=\nu (|z|+|z'|)\). Then \(Z\ge 2\nu \delta \), so that for \(\lambda \ge 1\), we have \(e^{-\lambda ^2Z}\le e^{-\nu \delta }e^{-\lambda ^2(Z-\nu \delta )}\), and integration with respect to \(\lambda \) yields

$$\begin{aligned} I(Z)\le e^{-\nu \delta }I(Z-\nu \delta )\le e^{-\nu \delta }I(\nu \delta ). \end{aligned}$$

By (A.2)–(A.3), we have

$$\begin{aligned} |k_\nu (x,z,x',z')|=\frac{4\nu ^4}{\pi C_F(\nu )}|K(X,Z)|\le \frac{4\nu ^4}{\pi C_F(\nu )}I(Z). \end{aligned}$$

Putting together these two estimates yields (A.13). \(\square \)

The following property of \(k_\nu \) will also prove useful.

Proposition A.4

For every \(v\in L^1(D)\), the function \(f_v\) defined by

$$\begin{aligned} f_v(x,z)=\int _Dk_\nu (x,z,x',z')v(x',z')dx'dz' \end{aligned}$$

is real analytic in \(\mathbb {R}\times \mathbb {R}^\star \).


Without loss of generality, we may assume \(\nu =1\) and \(z>0\). Using \(\cos (\lambda (x-x'))=\mathfrak {R}(e^{i\lambda (x-x')})\), we may write \(f_v=(4/\pi C_F(1))\mathfrak {R}(f_v^1+f_v^2)\) with

$$\begin{aligned} f_v^1(x,z)= & {} \int _D\left( \int _1^2e^{-\lambda ^2z}e^{i\lambda x}e^{-\lambda ^2|z'|}e^{-i\lambda x'}\frac{\lambda ^4}{\sqrt{\lambda ^2-1}}d\lambda \right) v(x',z')dx'dz'\\ f_v^2(x,z)= & {} \int _D\left( \int _2^{+\infty }e^{-\lambda ^2z}e^{i\lambda x}e^{-\lambda ^2|z'|}e^{-i\lambda x'}\frac{\lambda ^4}{\sqrt{\lambda ^2-1}}d\lambda \right) v(x',z')dx'dz'. \end{aligned}$$

It is sufficient to prove that \(f_v^1\) and \(f_v^2\) are real analytic on \(\mathbb {R}\times (0,+\infty )\) (with values in \(\mathbb {C}\) considered as a vector space over \(\mathbb {R}\)). For this purpose, it is sufficient to show (see, e.g., [6]) that for \(i=1\) or 2, \(f_v^i\) is \(C^\infty \) on \(\mathbb {R}\times (0,+\infty )\) and that for every compact subset K of \(\mathbb {R}^2\), there are positive constants \(C_K\) and \(M_K\) such that for all \(l=(l_1,l_2)\in \mathbb {N}^2\) and for all \((x,z)\in K\),

$$\begin{aligned} \left| \frac{\partial ^{l_1+l_2}f_v^i}{\partial x^{l_1}\partial z^{l_2}}(x,z)\right| \le C_KM_K^{|l|}|l|! \end{aligned}$$

where \(|l|=l_1+l_2\), as usual.

The function \(f_v^1\) is clearly of class \(C^\infty \) on \(\mathbb {R}\times (0,+\infty )\), and, for any \(l=(l_1,l_2)\in \mathbb {N}^2\),

$$\begin{aligned} \frac{\partial ^{l_1+l_2}f_v^1}{\partial x^{l_1}\partial z^{l_2}}(x,z)=\int _D\left( \int _1^2(-\lambda ^2)^{l_2}(i\lambda )^{l_1}e^{-\lambda ^2z}e^{i\lambda x}e^{-\lambda ^2|z'|}e^{-i\lambda x'}\frac{v(x',z')\lambda ^4}{\sqrt{\lambda ^2-1}}d\lambda \right) dx'dz. \end{aligned}$$

Thus, for \((x,z)\in \mathbb {R}\times (0,+\infty )\),

$$\begin{aligned} \left| \frac{\partial ^{l_1+l_2}f_v^1}{\partial x^{l_1}\partial z^{l_2}}(x,z)\right| \le 2^{l_1+2l_2+4}\Vert v\Vert _{L^1(D)}\int _1^2\frac{d\lambda }{\sqrt{\lambda ^2-1}}. \end{aligned}$$

Estimate (A.14) is satisfied with \(C_K=2^4\Vert v\Vert _{L^1(D)}\int _1^2(\lambda ^2-1)^{-1/2}d\lambda \) and \(M_K=4\).

Let now \(\delta \in (0,1)\). Then \(f_v^2\) is clearly of class \(C^\infty \) on \(\mathbb {R}\times (\delta ,+\infty )\); its partial derivatives are obtained on replacing the integral over (1, 2) in (A.15) by an integral over \((2,+\infty )\). This yields

$$\begin{aligned} \left| \frac{\partial ^{l_1+l_2}f_v^2}{\partial x^{l_1}\partial z^{l_2}}(x,z)\right| \le \Vert v\Vert _{L^1(D)}\int _2^{+\infty }\lambda ^{l_1+2l_2+4}e^{-\lambda ^2 z}\frac{d\lambda }{\sqrt{\lambda ^2-1}}. \end{aligned}$$

Next, we use that \((\lambda ^2-1)^{-1/2}\le C/\lambda \) on \((2,+\infty )\), for some constant C, and we perform the change of variable \(\mu =\lambda ^2 z\) in the integral. We obtain

$$\begin{aligned} \left| \frac{\partial ^{l_1+l_2}f_v^2}{\partial x^{l_1}\partial z^{l_2}}(x,z)\right| \le \frac{C\Vert v\Vert _{L^1(D)}}{2z^{l_1/2+l_2+2}}\Gamma (l_1/2+l_2+2), \end{aligned}$$

where \(\Gamma \) is the Gamma function. Next, we use that

$$\begin{aligned} \Gamma (l_1/2+l_2+2)\le \Gamma (l_1+l_2+2)=(l_1+l_2+2)(l_1+l_2)!. \end{aligned}$$

We find that estimate (A.14) is valid on \(\mathbb {R}\times (\delta ,+\infty )\) with, e.g.,

$$\begin{aligned} C_K=C'\Vert v\Vert _{L^1(D)}/(2\delta ^2),\quad C'=C\sup _{k\in \mathbb {N}} (k+2)/2^k,\quad \text{ and } \quad M_K=2/\delta . \end{aligned}$$

\(\square \)

Technical Lemmas

The proof of the following lemma may be found in [9].

Lemma B.1

Let \(B((x_0,z_0),r_0)\) be an open ball and \(U\in C^2(B((x_0,z_0),r_0))\). Then, for all \(r\in (0,r_0)\),

This remains valid for all \(U\in H^1(B((x_0,z_0),r_0))\) such that \(\Delta U\) is a measure satisfying

$$\begin{aligned} \int _0^rds\, s^{-1}\int _{B((x_0,z_0),s)}d|\Delta u|<\infty , \end{aligned}$$

and such that


Remark B.2

The proof shows furthermore that the condition (B.1) implies the existence of the limit in (B.2) for any \((x_0,z_0)\) whence we can take some precise representation of U defined thanks to (B.2).

The following lemma is more or less classical (see, e.g. [9, 17]).

Lemma B.3

Let \(B((x_0,z_0),r_0)\) be an open ball, \(r_0\le 1\), \(F\in L^p(B((x_0,z_0),r_0))\), \(p\in (1,2)\), \(\alpha =2/p'\). Then, there exists a constant C which depends only on p and \(\Vert F\Vert _{L^p(B((x_0,z_0),r_0))}\) such that, for \(r\in (0,r_0)\),

  1. (i)

    if \(\Delta U=F\) on \(B((x_0,z_0),r_0)\), then

    $$\begin{aligned} |U|_{\alpha ,B((x_0,z_0),r/2)}\le C\left[ 1+r^{-\alpha }\Vert U\Vert _{L^\infty (B((x_0,z_0),r))}\right] , \end{aligned}$$
  2. (ii)

    if \(\Delta U\ge F\) and \(U\ge 0\) on \(B((x_0,z_0),r_0)\), then



Recall that for the solution of

$$\begin{aligned} W\in H^1_0(B_1),\quad -\Delta W=G \text{ on } B_1, \end{aligned}$$

since \(p>1\), by elliptic regularity we have

$$\begin{aligned} \Vert W\Vert _{W^{2,p}(B_1)}\le C(p)\Vert G\Vert _{L^p(B_1)}. \end{aligned}$$

We use the Sobolev imbedding \(W^{2,p}(B_1)\subset C^\alpha (\overline{B_1})\) [7] and we apply this to the rescaled functions

$$\begin{aligned} \forall \xi \in B_1,\quad V(\xi ,\zeta )=U((x_0,z_0)+r(\xi ,\zeta )),\ G(\xi ,\zeta )=r^2F((x_0,z_0)+r(\xi ,\zeta )). \end{aligned}$$

We obtain

$$\begin{aligned} \Vert W\Vert _{L^\infty (B_1)}+|W|_{\alpha ,B_1}\le C'(p) r^{2-2/p}\Vert F\Vert _{L^p(B((x_0,z_0),r_0))}. \end{aligned}$$

For (B.3), we notice that \(\Delta (V-W)=0\) on \(B_1\) so that by Harnack’s inequality [17],

$$\begin{aligned} |V-W|_{\alpha ,B_{1/2}}\le \Vert \nabla (V-W)\Vert _{L^\infty (B_{1/2})}\le C\Vert V\Vert _{L^\infty (\partial B_1)}. \end{aligned}$$

Together with (B.5), this inequality gives

$$\begin{aligned} |V|_{\alpha ,B_{1/2}}\le C(p,\Vert F\Vert _{L^p(B((x_0,z_0),r_0))})\left[ r^\alpha +\Vert V\Vert _{L^\infty (\partial B_1)}\right] . \end{aligned}$$

Going back to U gives (B.3) by change of variable. For (B.4), we first notice that \(-\Delta (V-W)\le 0\), so that \((V-W)(x,z)\le \int _{\partial B_1}P_{(x,z)}(x',z')V(x',z')d\sigma (x',z')\) where \(P_{x,z}(\cdot )\) denotes the Poisson kernel at (xz). Using (B.5) again and \(V\ge 0\), we deduce that

The relation (B.4) follows by change of variable. \(\square \)

The following lemma is proved in [9].

Lemma B.4

Let \(B((x_0,z_0),r_0)\) be an open ball, \(r_0\le 1\), \(F\in L^q(B((x_0,z_0),r_0))\), \(q>2\). Then, there exists a constant \(C=C(q,\Vert F\Vert _{L^q(B((x_0,z_0),r_0))})\) such that, for \(r\in (0,r_0)\),

  1. (i)

    if \(\Delta U=F\) on \(B((x_0,z_0),r_0)\), then

    $$\begin{aligned} |U|_{1,B((x_0,z_0),r/2)}\le C\left[ 1+r^{-1}\Vert U\Vert _{L^\infty (B((x_0,z_0),r))}\right] , \end{aligned}$$
  2. (ii)

    if \(\Delta U\ge F\) and \(U\ge 0\) on \(B((x_0,z_0),r_0)\), then

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Dambrine, J., Pierre, M. Regularity of Optimal Ship Forms Based on Michell’s Wave Resistance. Appl Math Optim 82, 23–62 (2020).

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  • Shape optimization
  • Existence
  • Regularity
  • Dirichlet energy
  • Wave resistance

Mathematics Subject Classification

  • 49N60
  • 76B75
  • 49J40
  • 49Q10