Abstract.
We prove local Lipschitz continuity of the solution to the state equation in two kinds of shape optimization problems with constraint on the volume: the minimal shaping for the Dirichlet energy, with no sign condition on the state function, and the minimal shaping for the first eigenvalue of the Laplacian. This is a main first step for proving regularity of the optimal shapes themselves.
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Aguilera, N., Alt, H.W., Caffarelli, L.A.: An optimization problem with volume constraint. SIAM J. Control Optimization 24, 191-198 (1986)
Alt, H.W., Caffarelli, L.A.: Existence and regularity for a minimum problem with free boundary. J. reine angew. Math. 325, 105-144 (1981)
Alt, H.W., Caffarelli, L.A., Friedman, A.: Variational problems with two phases and their free boundaries. Trans. AMS 282(2), 431-461 (1984)
Briançon, T.: Regularity of optimal shapes for the Dirichlet’s energy with volume constraint. ESAIM: COCV 10, 99-122 (2004)
Buttazzo, G., Dal Maso, G.: An existence result for a class of Shape Optimization Problems. Arch. Ration. Mech. Anal. 122, 183-195 (1993)
Caffarelli, L.A., Jerison, D., Kenig, C.E.: Some new monotonicity theorems with applications to free boundary problems. Annals of Math. 155, 369-404 (2002)
Crouzeix, M.: Variational approach of a magnetic shaping problem. Eur. J. Mech. B Fluids 10, 527-536 (1991)
Descloux, J.: A stability result for the magnetic shaping problem. Z. Angew. Math. Phys. 45(4): 543-555 (1994)
Descloux, J.: Stability of the solutions of the bidimensional magnetic shaping problem in absence of surface tension. Eur. J. Mech, B/Fluids 10, 513-526 (1991)
Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL 1992
Friedman, A., Liu, Y.: A free boundary problem arising in magnetohydrodynamic system. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22(4), 375-448 (1995)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer, Berlin 1983
Giusti, E.: Minimal surfaces and functions of bounded variation. Monographs in Mathematics, 80, Birkhäuser, Basel Boston, MA 1984
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)
Hayouni, M.: Lipschitz continuity of the state function in a shape optimization problem. J. Conv. Anal. 6, 71-90 (1999)
Hayouni, M.: Sur la minimisation de la premiére valeur propre du laplacien. CRAS 330 7, 551-556 (2000)
Hedberg, L.I.: Spectral synthesis in Sobolev space and uniqueness of solutions of the Dirichlet problem. Acta Math. 147, 237-264 (1981)
Heinonen, J., Kipelanen, T., Martio, O.: Nonlinear potential theory of degenerate elliptic equations. Calderon Press, Oxford 1993
Henrot, A., Pierre, M.: Un probléme inverse en formage de métaux liquides. RAIRO Modél. Math. Anal. Num. 23, 155-177 (1989)
Morrey, C.B.: Multiple integrals in the calculus of variations. Sringer, Berlin, Heidelberg New York 1966
Tilli, P.: On a constrained variational problem with an arbitrary number of free boundaries. Interfaces Free Bound. 2(2), 201-212 (2000)
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Received: 11 November 2003, Accepted: 10 May 2004, Published online: 8 February 2005
Mathematics Subject Classification (2000):
49Q10, 35R35, 49K20, 35J20
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Briançon, T., Hayouni, M. & Pierre, M. Lipschitz continuity of state functions in some optimal shaping. Calc. Var. 23, 13–32 (2005). https://doi.org/10.1007/s00526-004-0286-5
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DOI: https://doi.org/10.1007/s00526-004-0286-5