# Regularity results for an optimal design problem with a volume constraint

Menita Carozza; Irene Fonseca; Antonia Passarelli di Napoli

ESAIM: Control, Optimisation and Calculus of Variations (2014)

- Volume: 20, Issue: 2, page 460-487
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topCarozza, Menita, Fonseca, Irene, and Passarelli di Napoli, Antonia. "Regularity results for an optimal design problem with a volume constraint." ESAIM: Control, Optimisation and Calculus of Variations 20.2 (2014): 460-487. <http://eudml.org/doc/272789>.

@article{Carozza2014,

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.},

author = {Carozza, Menita, Fonseca, Irene, Passarelli di Napoli, Antonia},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {regularity; nonlinear variational problem; free interfaces},

language = {eng},

number = {2},

pages = {460-487},

publisher = {EDP-Sciences},

title = {Regularity results for an optimal design problem with a volume constraint},

url = {http://eudml.org/doc/272789},

volume = {20},

year = {2014},

}

TY - JOUR

AU - Carozza, Menita

AU - Fonseca, Irene

AU - Passarelli di Napoli, Antonia

TI - Regularity results for an optimal design problem with a volume constraint

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2014

PB - EDP-Sciences

VL - 20

IS - 2

SP - 460

EP - 487

AB - 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.

LA - eng

KW - regularity; nonlinear variational problem; free interfaces

UR - http://eudml.org/doc/272789

ER -

## References

top- [1] H.W. Alt and L.A. Caffarelli, Existence and regularity results for a minimum problem with free boundary. J. Reine Angew. Math.325 (1981) 107–144. Zbl0449.35105MR618549
- [2] E. Acerbi and N. Fusco, Regularity for minimizers of non-quadratic functionals: the case 1 < p < 2. J. Math. Anal. Appl. 140 (1989) 115–135. Zbl0686.49004MR997847
- [3] E. Acerbi and N. Fusco, A regularity theorem for minimizers of quasi-convex integrals. Arch. Rational Mech. Anal.99 (1987) 261–281. Zbl0627.49007MR888453
- [4] L. Ambrosio and G. Buttazzo, An optimal design problem with perimeter penalization. Calc. Var. Partial Differ. Eq.1 (1993) 55–69. Zbl0794.49040MR1261717
- [5] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). Zbl0957.49001MR1857292
- [6] E. Bombieri, Regularity theory for almost minimal currents. Arch. Rational Mech. Anal.78 (1982) 99–130. Zbl0485.49024MR648941
- [7] M. Carozza and A. Passarelli Di Napoli, A regularity theorem for minimisers of quasiconvex integrals: The case 1 < p < 2. Proc. Roy. Soc. Edinburgh A Math. 126, 6 (1996) 1181–1200. Zbl0955.49021MR1424221
- [8] L. Esposito and N. Fusco, A remark on a free interface problem with volume constraint. J. Convex Anal.18 (2011) 417–426. Zbl1216.49033MR2828497
- [9] L.C. Evans and F.R. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992). Zbl0804.28001MR1158660
- [10] I. Fonseca and N. Fusco, Regularity results for anisotropic image segmentation models. Ann. Sci. Norm. Super. Pisa24 (1997) 463–499. Zbl0899.49018MR1612389
- [11] I. Fonseca, N. Fusco, G. Leoni and V. Millot, Material voids in elastic solids with anisotropic surface energies. J. Math. Pures Appl. 96 (2011). Zbl1285.74003MR2851683
- [12] I. Fonseca, N. Fusco, G. Leoni and M. Morini, Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results. Arch. Rational Mech. Anal.186 (2007) 477–537. Zbl1126.74029MR2350364
- [13] N. Fusco and J. Hutchinson, C1,α partial regularity of functions minimising quasiconvex integrals. Manuscripta Math.54 (1985) 121–143. Zbl0587.49005MR808684
- [14] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear ellyptic systems. Ann. Math. Stud. Princeton University Press (1983). Zbl0516.49003MR717034
- [15] M. Giaquinta and G. Modica, Partial regularity of minimizers of quasiconvex integrals. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3 (1986) 185–208. Zbl0594.49004MR847306
- [16] 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). Zbl0562.35001MR737190
- [17] E. Giusti. Direct methods in the calculus of variations. World Scientific (2003). Zbl1028.49001MR1962933
- [18] M. Gurtin, On phase transitions with bulk, interfacial, and boundary energy. Arch. Rational Mech. Anal.96 (1986) 243–264 MR855305
- [19] C.J. Larsen, Regularity of components in optimal design problems with perimeter penalization. Calc. Var. Partial Differ. Eq.16 (2003) 17–29. Zbl1083.49031MR1951490
- [20] F.H. Lin, Variational problems with free interfaces. Calc. Var. Partial Differ. Eq.1 (1993) 149–168. Zbl0794.49038MR1261721
- [21] F.H. Lin and R.V. Kohn, Partial regularity for optimal design problems involving both bulk and surface energies. Chin. Ann. Math. B 20, (1999) 137–158. Zbl0946.49034MR1699139
- [22] V. Šverák and X. Yan. Non-Lipschitz minimizers of smooth uniformly convex variational integrals. Proc. Natl. Acad. Sci. USA99 (2002) 15269–15276. Zbl1106.49046MR1946762
- [23] I. Tamanini, Boundaries of Caccioppoli sets with Hölder-continuous normal vector. J. Reine Angew. Math.334 (1982) 27–39. Zbl0479.49028MR667448

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.