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Regularity results for an optimal design problem with a volume constraint

Published online by Cambridge University Press:  07 March 2014

Menita Carozza ,
Irene Fonseca  and
Antonia Passarelli di Napoli
Menita Carozza
Affiliation:
Dipartimento di Ingegneria – Università del Sannio, Corso Garibaldi 82100 Benevento, Italy. [email protected]
Irene Fonseca
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, PA 15213-3890 Pittsburgh, USA; [email protected]
Antonia Passarelli di Napoli
Affiliation:
Università di Napoli “Federico II” Dipartimento di Mat. e Appl. ‘R. Caccioppoli’, Via Cintia, 80126 Napoli, Italy; [email protected]
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Abstract

Regularity results for minimal configurations of variational problems involving both bulk and surface energies and subject to a volume constraint are established. The bulk energies are convex functions with p-power growth, but are otherwise not subjected to any further structure conditions. For a minimal configuration (u,E), Hölder continuity of the function u is proved as well as partial regularity of the boundary of the minimal set E. Moreover, full regularity of the boundary of the minimal set is obtained under suitable closeness assumptions on the eigenvalues of the bulk energies.

Keywords

Regularitynonlinear variational problemfree interfaces
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 2 , April 2014 , pp. 460 - 487
Copyright
© EDP Sciences, SMAI, 2014

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References

Alt, H.W. and Caffarelli, L.A., Existence and regularity results for a minimum problem with free boundary. J. Reine Angew. Math. 325 (1981) 107–144. Google Scholar
Acerbi, E. and Fusco, N., Regularity for minimizers of non-quadratic functionals: the case 1 < p < 2. J. Math. Anal. Appl. 140 (1989) 115–135. Google Scholar
Acerbi, E. and Fusco, N., A regularity theorem for minimizers of quasi-convex integrals. Arch. Rational Mech. Anal. 99 (1987) 261–281. Google Scholar
Ambrosio, L. and Buttazzo, G., An optimal design problem with perimeter penalization. Calc. Var. Partial Differ. Eq. 1 (1993) 55–69. Google Scholar
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000).
Bombieri, E., Regularity theory for almost minimal currents. Arch. Rational Mech. Anal. 78 (1982) 99–130. Google Scholar
Carozza, M. and Passarelli Di Napoli, A., A regularity theorem for minimisers of quasiconvex integrals: The case 1 < p < 2. Proc. Roy. Soc. Edinburgh A Math. 126, 6 (1996) 1181–1200. Google Scholar
Esposito, L. and Fusco, N., A remark on a free interface problem with volume constraint. J. Convex Anal. 18 (2011) 417–426. Google Scholar
L.C. Evans and F.R. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992).
Fonseca, I. and Fusco, N., Regularity results for anisotropic image segmentation models. Ann. Sci. Norm. Super. Pisa 24 (1997) 463–499. Google Scholar
Fonseca, I., Fusco, N., Leoni, G. and Millot, V., Material voids in elastic solids with anisotropic surface energies. J. Math. Pures Appl. 96 (2011). Google Scholar
Fonseca, I., Fusco, N., Leoni, G. and Morini, M., Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results. Arch. Rational Mech. Anal. 186 (2007) 477–537. Google Scholar
Fusco, N. and Hutchinson, J., C 1,α partial regularity of functions minimising quasiconvex integrals. Manuscripta Math. 54 (1985) 121–143. Google Scholar
M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear ellyptic systems. Ann. Math. Stud. Princeton University Press (1983).
Giaquinta, M. and Modica, G., Partial regularity of minimizers of quasiconvex integrals. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3 (1986) 185–208. Google Scholar
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, 2nd edn., vol. 224 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin (1983).
E. Giusti. Direct methods in the calculus of variations. World Scientific (2003).
Gurtin, M., On phase transitions with bulk, interfacial, and boundary energy. Arch. Rational Mech. Anal. 96 (1986) 243–264 Google Scholar
Larsen, C.J., Regularity of components in optimal design problems with perimeter penalization. Calc. Var. Partial Differ. Eq. 16 (2003) 17–29. Google Scholar
Lin, F.H., Variational problems with free interfaces. Calc. Var. Partial Differ. Eq. 1 (1993) 149–168. Google Scholar
Lin, F.H. and Kohn, R.V., Partial regularity for optimal design problems involving both bulk and surface energies. Chin. Ann. Math. B 20, (1999) 137–158. Google Scholar
Šverák, V. and Yan, X.. Non-Lipschitz minimizers of smooth uniformly convex variational integrals. Proc. Natl. Acad. Sci. USA 99 (2002) 15269–15276. Google Scholar
Tamanini, I., Boundaries of Caccioppoli sets with Hölder-continuous normal vector. J. Reine Angew. Math. 334 (1982) 27–39. Google Scholar