Metadata
Abstract
We consider shape optimization problems for general integral functionals of the calculus of variations, defined on a domain that varies over all subdomains of a given bounded domain of . We show in a rather elementary way the existence of a solution that is in general a quasi open set. Under very mild conditions we show that the optimal domain is actually open and with finite perimeter. Some counterexamples show that in general this does not occur.
References
[BB05] Variational Methods in Shape Optimization Problems, Progress in Nonlinear Differential Equations, Volume 65, Birkhäuser, 2005 | MR | Zbl
[BDM93] An existence result for a class of shape optimization problems, Arch. Ration. Mech. Anal., Volume 122 (1993) no. 2, pp. 183-195 | DOI | MR | Zbl
[BHP05] Lipschitz continuity of state functions in some optimal shaping, Calc. Var. Partial Differ. Equ., Volume 23 (2005) no. 1, pp. 13-32 | DOI | MR | Zbl
[Bri04] Regularity of optimal shapes for the DirichletÕs energy with volume constraint, ESAIM, Control Optim. Calc. Var., Volume 10 (2004) no. 1, pp. 99-122 | DOI | MR | Zbl
[Buc12] Minimization of the -th eigenvalue of the Dirichlet Laplacian, Arch. Ration. Mech. Anal., Volume 206 (2012) no. 3, pp. 1073-1083 | DOI | MR | Zbl
[But89] Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Reseqrch Notes in Mathematics Series, Volume 207, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989 | MR | Zbl
[GG84] Quasi-minima, Ann. Inst. Henri Poincaré Anal. Non Linéaire, Volume 1 (1984) no. 2, pp. 7-107 | Numdam | Zbl
[Giu03] Direct Methods in the Calculus of Variations, World Scientific Publishing, Singapore, 2003 | Zbl
[Hay99] Lipschitz continuity of the state function in a shape optimization problem, J. Conv. Anal., Volume 6 (1999) no. 1, pp. 71-90 | MR | Zbl
[HM05] Variation et Optimisation de Formes. Une Analyse Géométrique, Mathématiques & Applications, Volume 48, Springer, 2005 | Zbl
[Maz11] Sobolev Spaces with applications to elliptic partial differential equations, Grundlehren der Mathematischen Wissenschaften, Volume 342, Springer, 2011 | MR | Zbl
[Zie89] Weakly Differentiable Functions, Springer, 1989 | Zbl