Analysis and numerical studies of a problem of shape design. (English) Zbl 0726.65071

The paper deals with the following problem of shape design: (P) minimize \(J(v)=\int_{\Omega}\{g(| \nabla v|)-v\}dx\) on \(W_ 0^{1,2}(\Omega)\), where \(\Omega \subset {\mathbb{R}}^ N\) is a bounded simply connected domain, \(N\geq 2\). The uniqueness for smooth solutions to (P) is studied and some numerical methods are applied to verify certain analytical results.


65K10 Numerical optimization and variational techniques
49M27 Decomposition methods
49J40 Variational inequalities
Full Text: DOI


[1] Alexandrov, A. D., Uniqueness theorems for surfaces in the large, part V. Vestnik Leningrad University 13 (1958) p. 5-8.
[2] Bauman, P., & D. Phillips, A nonconvex variational problem related to change of phase. Appl. Math. Optim. 21 (1990) p. 113-138. · Zbl 0686.73018
[3] Buononcore, P., Some isoperimetric inequalities in a special case of the problem of torsional creep. Applic. Anal. 27 (1988) pp. 133-142. · Zbl 0617.35037
[4] Chipot, M., & L. C. Evans, Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations. Proc. Roy. Soc. Edinburgh 102 A (1986) pp. 291-303. · Zbl 0602.49029
[5] Dacorogna, B., Direct methods in the calculus of variations. Springer (1989) Berlin. · Zbl 0703.49001
[6] French, D., On the convergence of finite element approximations of relaxed variational problem. IMA Preprint 503, Minneapolis 1989.
[7] Giaquinta, M., & E. Giusti, On the regularity of the minima of variational integrals. Acta Math. 148 (1982) pp. 31-46. · Zbl 0494.49031
[8] Goodman, J., Kohn, V. R., & L. Reyna, Numerical study of a relaxed variational problem from optimal design. Computer Methods in Applied Math. and Engin. 57 (1986) pp. 107-127. · Zbl 0591.73119
[9] Hackbusch, W., Multi-grid methods. Springer-Verlag (1985) Heidelberg. · Zbl 0595.65106
[10] Hackbusch, W., Theorie und Numerik elliptischer Differentialgleichungen. Teubner-Verlag (1986) Stuttgart. · Zbl 0609.65065
[11] Hackbusch, W., & A. Reusken, Analysis of a damped nonlinear multilevel method. Numer. Math. 55 (1989) pp. 225-246. · Zbl 0673.65031
[12] Hoppe, R., & R. Kornhuber, Multi-grid methods for the two phase Stefan problem. Report 171, Techn. Univ. Berlin (1987). · Zbl 0655.65137
[13] Kawohl, B., Rearrangements and convexity of level sets in PDE. Springer Lecture Notes in Math. 1150 (1985) Heidelberg. · Zbl 0593.35002
[14] Kohn, R., & G. Strang, Optimal design and relaxation of variational problems I, II, III. Comm. Pure Appl. Math. 39 (1986) pp. 113-137, 139-182, 353-377. · Zbl 0609.49008
[15] Lurie, K. A., A. V. Cherkaev & A. V. Fedorov, Regularization of optimal design problems for bars and plates. J. Optim. Theory Appl. 37 (1982) pp. 499-543. · Zbl 0464.73109
[16] Murat, F., & L. Tartar, Optimality conditions and homogenization, in: Nonlinear Variational Problems. Eds.: A. Marino, L. Modica, S. Spagnolo & M. Degiovanni, Pitman Research Notes in Math. 127 (1985) pp. 1-8. · Zbl 0569.49015
[17] Murat, F., & L. Tartar, Calcul des variations et homogenization, in: Les méthodes de l’homogeneisation: théorie et applications en physique. Eds.: D. Bergman et al. Collection de la Direction des Études et Recherches d’Electricité de France 57 (1985) pp. 319-369.
[18] Talenti, G., Non linear elliptic equations, rearrangements of functions and Orlicz spaces. Ann. Mat. Pura Appl. Ser. IV 120 (1977) pp. 159-184. · Zbl 0419.35041
[19] Voas, C., & D. Yaniro, Symmetrization and optimal control for elliptic equations. Proc. Amer. Math. Soc. 99 (1987) pp. 509-514. · Zbl 0609.49016
[20] Wittum, G., Linear iterations as smoothers in multigrid methods: Theory with applications to incomplete decompositions. Impact of Computing in Science and Engineering 1 (1989) pp. 180-215. · Zbl 0707.65020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.