Optimal design via variational principles: The three-dimensional case. (English) Zbl 1038.49020

Bellido, José Carlos; Pedregal, Pablo

J. Math. Anal. Appl. 287, No. 1, 157-176 (2003).

The paper aims to start a systematic analysis of some three-dimensional optimization problems, generalizing some results of the same authors.
The problems taken into account deal with the minimization of the integral \[ I(y,u)=\int_\Omega F(x,y(x),u(x),\nabla u(x))\,dx, \] subject to \(y\in L^\infty(\Omega;{\mathbb R}^n)\), \(y(x)\in K\) for a.e. \(x\in\Omega\), \(\text{ div}(G(x,y(x),u(x),\nabla u(x)))=0\) in \(\Omega\), boundary conditions on \(u\), and \(\int_\Omega H(x,y(x),u(x),\nabla u(x))\,dx\leq\alpha\). Here \(\Omega\subseteq{\mathbb R}^3\) is a regular domain, and \(K\) is a closed subset of \({\mathbb R}^n\). By the introduction of a suitable vector field \(V\), the problem is transformed into the minimization of \[ \widetilde I(u,V)=\int_\Omega \widetilde F(x,u(x),\nabla u(x),V(x))dx, \] subject to \(\text{div\,}V=0\) in \(\Omega\), and boundary conditions on \(u\). Hypotheses for the new variational problem are derived in order to provide existence results. Nonexistence and relaxation properties are also briefly analyzed.
The underlying Young measures associated with the above divergence free constraint are also considered.

Reviewer: Riccardo De Arcangelis (Napoli)

MSC:

49J45	Methods involving semicontinuity and convergence; relaxation
49J10	Existence theories for free problems in two or more independent variables

Keywords:

divergence-free constraints; existence; optimization; relaxation; Young measures

References:

[1]	Bellido, J. C.; Pedregal, P., Optimal design via variational principles: the one-dimensional case, J. Math. Pures Appl., 80, 245-261 (2001) · Zbl 0985.49006
[2]	Bellido, J. C.; Pedregal, P., Optimal design via variational principles: the two-dimensional case, J. Convex Anal. (2003), in press · Zbl 1038.49020
[3]	Bellido, J. C., Explicit quasiconvexification of a relaxed density coming from a three-dimensional optimal design problem, Nonlinear Anal., 52, 1709-1726 (2003) · Zbl 1016.49015
[4]	Dacorogna, B., Weak Continuity and Weak Lower Semicontinuity for Non-Linear Functionals. Weak Continuity and Weak Lower Semicontinuity for Non-Linear Functionals, Springer Lecture Notes, 922 (1982) · Zbl 0484.46041
[5]	Dacorogna, B., Direct Methods in the Calculus of Variations (1989), Springer · Zbl 0703.49001
[6]	Fonseca, I.; Müller, S., \(A\)-quasiconvexity, lower semicontinuity and Young measures, SIAM Math. Anal., 30, 1355-1390 (1999) · Zbl 0940.49014
[7]	Girault, V.; Raviart, P. A., Finite Element Methods for Navier-Stokes Equations (1986), Springer: Springer Berlin · Zbl 0396.65070
[8]	Kohn, R., The relaxation of a double-well energy, Contin. Mech. Thermodyn., 3, 193-236 (1991) · Zbl 0825.73029
[9]	Kohn, R. V.; Vogelius, M., Relaxation of a variational method for impedance computed tomography, Comm. Pure Appl. Math., 40, 745-777 (1987) · Zbl 0659.49009
[10]	Matousek, J.; Plechac, P., On functional separately convex hulls, Discrete Comput. Geom., 19, 105-130 (1998) · Zbl 0892.68102
[11]	Murat, F., Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. (IV), 5, 489-507 (1978) · Zbl 0399.46022
[12]	Murat, F., Compacité par compensation II, (De Giorgi, E.; Magenes, E.; Mosco, U., Recent Methods in Nonlinear Analysis, Proceedings (1979), Pitagora: Pitagora Bologna), 245-256 · Zbl 0427.35008
[13]	Pedregal, P., Parametrized Measures and Variational Principles (1997), Birkhäuser: Birkhäuser Basel · Zbl 0879.49017
[14]	Pedregal, P., Optimal design and constrained quasiconvexity, SIAM J. Math. Anal., 32, 854-869 (2000) · Zbl 0995.49004
[15]	Pedregal, P., Constrained quasiconvexity and structural optimization, Arch. Rational Mech. Anal., 154, 325-342 (2000) · Zbl 0970.74053
[16]	Pedregal, P., Fully explicit quasiconvexification of the mean-square deviation of the gradient of the state in optimal design, Electron. Res. Announc. Amer. Math. Soc., 7, 72-78 (2001) · Zbl 0980.49021
[17]	Serre, D., Formes quadratiques et calcul des variations, J. Math. Pures Appl., 62, 177-196 (1983) · Zbl 0529.49005
[18]	Tartar, L., Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics, (Knops, R., Heriot-Watt Symposium, Vol. IV. Heriot-Watt Symposium, Vol. IV, Pitman Res. Notes Math., 39 (1979)), 136-212 · Zbl 0437.35004
[19]	Tartar, L., The compensated compactness method applied to systems of conservation laws, (Ball, J. M., Systems of Nonlinear Partial Differential Equations (1983), Riedel), 168-187