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: regularity, nonlinear variational problem, free interfaces

@article{COCV_2014__20_2_460_0, author = {Carozza, Menita and Fonseca, Irene and Passarelli di Napoli, Antonia}, title = {Regularity results for an optimal design problem with a volume constraint}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {460--487}, publisher = {EDP-Sciences}, volume = {20}, number = {2}, year = {2014}, doi = {10.1051/cocv/2013071}, mrnumber = {3264212}, zbl = {1286.49041}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2013071/} }

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 SP - 460 EP - 487 VL - 20 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2013071/ DO - 10.1051/cocv/2013071 LA - en ID - COCV_2014__20_2_460_0 ER -

%0 Journal Article %A Carozza, Menita %A Fonseca, Irene %A Passarelli di Napoli, Antonia %T Regularity results for an optimal design problem with a volume constraint %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 460-487 %V 20 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2013071/ %R 10.1051/cocv/2013071 %G en %F COCV_2014__20_2_460_0

Carozza, Menita; Fonseca, Irene; Passarelli di Napoli, Antonia. Regularity results for an optimal design problem with a volume constraint. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 2, pp. 460-487. doi : 10.1051/cocv/2013071. http://archive.numdam.org/articles/10.1051/cocv/2013071/

[1] Existence and regularity results for a minimum problem with free boundary. J. Reine Angew. Math. 325 (1981) 107-144. | MR | Zbl

and ,[2] Regularity for minimizers of non-quadratic functionals: the case 1 < p < 2. J. Math. Anal. Appl. 140 (1989) 115-135. | MR | Zbl

and ,[3] A regularity theorem for minimizers of quasi-convex integrals. Arch. Rational Mech. Anal. 99 (1987) 261-281. | MR | Zbl

and ,[4] An optimal design problem with perimeter penalization. Calc. Var. Partial Differ. Eq. 1 (1993) 55-69. | MR | Zbl

and ,[5] Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). | MR | Zbl

, and ,[6] Regularity theory for almost minimal currents. Arch. Rational Mech. Anal. 78 (1982) 99-130. | MR | Zbl

,[7] A regularity theorem for minimisers of quasiconvex integrals: The case 1 < p < 2. Proc. Roy. Soc. Edinburgh A Math. 126, 6 (1996) 1181-1200. | MR | Zbl

and ,[8] A remark on a free interface problem with volume constraint. J. Convex Anal. 18 (2011) 417-426. | MR | Zbl

and ,[9] Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992). | MR | Zbl

and ,[10] Regularity results for anisotropic image segmentation models. Ann. Sci. Norm. Super. Pisa 24 (1997) 463-499. | Numdam | MR | Zbl

and ,[11] Material voids in elastic solids with anisotropic surface energies. J. Math. Pures Appl. 96 (2011). | MR | Zbl

, , and ,[12] Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results. Arch. Rational Mech. Anal. 186 (2007) 477-537. | MR | Zbl

, , and ,[13] C1,α partial regularity of functions minimising quasiconvex integrals. Manuscripta Math. 54 (1985) 121-143. | MR | Zbl

and ,[14] Multiple integrals in the calculus of variations and nonlinear ellyptic systems. Ann. Math. Stud. Princeton University Press (1983). | MR | Zbl

,[15] Partial regularity of minimizers of quasiconvex integrals. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3 (1986) 185-208. | Numdam | MR | Zbl

and ,[16] Elliptic partial differential equations of second order, 2nd edn., vol. 224 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin (1983). | MR | Zbl

and ,[17] Direct methods in the calculus of variations. World Scientific (2003). | MR | Zbl

.[18] On phase transitions with bulk, interfacial, and boundary energy. Arch. Rational Mech. Anal. 96 (1986) 243-264 | MR

,[19] Regularity of components in optimal design problems with perimeter penalization. Calc. Var. Partial Differ. Eq. 16 (2003) 17-29. | MR | Zbl

,[20] Variational problems with free interfaces. Calc. Var. Partial Differ. Eq. 1 (1993) 149-168. | MR | Zbl

,[21] Partial regularity for optimal design problems involving both bulk and surface energies. Chin. Ann. Math. B 20, (1999) 137-158. | MR | Zbl

and ,[22] Non-Lipschitz minimizers of smooth uniformly convex variational integrals. Proc. Natl. Acad. Sci. USA 99 (2002) 15269-15276. | MR | Zbl

and .[23] Boundaries of Caccioppoli sets with Hölder-continuous normal vector. J. Reine Angew. Math. 334 (1982) 27-39. | MR | Zbl

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