zbMATH — the first resource for mathematics

Existence theorem for a minimum problem with free discontinuity set. (English) Zbl 0682.49002
Summary: We study the variational problem \[ \min \{\int_{\Omega \setminus K}| \nabla u|^ 2dx+\mu \int_{\Omega \setminus K}| u- g|^ qdx+\lambda H_{n-1}(K\cap \Omega);\quad K\subset {\mathbb{R}}^ n\quad closed\quad set,\quad u\in C^ 1(\Omega \setminus K)\}, \] where \(\Omega\) is an open set in \({\mathbb{R}}^ n\), \(n\geq 2\), \(g\in L^ q(\Omega)\cap L^{\infty}(\Omega)\), \(1\leq q<+\infty\), \(0<\lambda\), \(\mu <+\infty\) and \(H_{n-1}\) is the (n-1)-dimensional Hausdorff measure.

49J10 Existence theories for free problems in two or more independent variables
49J45 Methods involving semicontinuity and convergence; relaxation
Full Text: DOI
[1] S. Agmon, A. Douglis &L. Nirenberg:Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary value conditions I, Comm. Pure Appl. Math.12, (1959), 623-727. · Zbl 0093.10401
[2] L. Ambrosio:A compactness theorem for a special class of functions of bounded variation, Preprint SNS, Pisa 1988, to appear in Boll. Un. Mat. Ital.
[3] L. Ambrosio:Existence theory for a new class of variational problems, Preprint SNS, Pisa 1988, to appear in Arch. Rational Mech. Anal. · Zbl 0663.49017
[4] J. Blat & J. M. Morel:Elliptic problems in image segmentation, Preprint 1988.
[5] H. Brezis:Analyse Fonctionnelle, Théorie et Applications. Masson, Paris, 1983.
[6] H. Brezis, J. M. Coron &E. H. Lieb:Harmonic maps with defects, Comm. Math. Phys.107 (1986), 679-705. · Zbl 0608.58016
[7] M. Carriero, A. Leaci, D. Pallara &E. Pascali:Euler conditions for a minimum problem with free discontinuity surfaces, Preprint Dipartimento di Matematica, Lecce, 1988.
[8] S. Chandrasekhar:Liquid Crystals. Cambridge University Press, Cambridge, 1977.
[9] E. De Giorgi:Nuovi teoremi relativi alle misure (r-1)-dimensionali in uno spazio ad r dimensioni, Ricerche Mat.4 (1955), 95-113. · Zbl 0066.29903
[10] E. De Giorgi:Free discontinuity problems in Calculus of Variations, Proceedings of the Meeting in honor ofJ. L. Lions, Paris 1988, to appear. · Zbl 0758.49002
[11] E. De Giorgi & L. Ambrosio:Un nuovo tipo di funzionale del calcolo delle variazioni, Atti Accad. Naz. Lincei, to appear.
[12] E. De Giorgi, F. Colombini &L. C. Piccinini:Frontiere Orientate di Misura Minima e Questioni Collegate, Quaderni S.N.S., Editrice Tecnico Scientifica, Pisa, 1972.
[13] E. De Giorgi, G. Congedo & I. Tamanini:Problemi di regolarità per un nuovo tipo di funzionale del calcolo delle variazioni, Atti Accad, Naz. Lincei, to appear.
[14] J. L. Ericksen,Equilibrium Theory of Liquid Crystals, Adv. Liq. Cryst., Vol. 2,G. H. Brown, Ed., Academic Press, New York, 1976, 233-299.
[15] H. Federer:Geometric Measure Theory. Springer-Verlag, Berlin, 1969. · Zbl 0176.00801
[16] H. Federer &W. P. Ziemer:The Lebesgue set of a function whose distributions derivatives are p-th power summable, Indiana Univ. Math. J.22 (1972), 139-158. · Zbl 0238.28015
[17] E. Giusti:Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston, 1984. · Zbl 0545.49018
[18] U. Massari &M. Miranda:Minimal Surfaces of Codimension One. North-Holland, Amsterdam, 1984. · Zbl 0565.49030
[19] M. Miranda:Distribuzioni aventi derivate misure, insiemi di perimetro localmente finito, Ann. Scuola Norm. Sup. Pisa18 (1964), 27-56. · Zbl 0131.11802
[20] J. M. Morel &S. Solimini:Segmentation of images by variational methods (on a model of Mumford and Shah), Revista Matem. de la Univ. Complutense de Madrid1 (1988), 169-182. · Zbl 0679.68205
[21] D. Mumford & J. Shah:Optimal approximations by piecewise smooth functions and associated variational problems, Preprint, to appear in Comm. Pure Appl: Math. · Zbl 0691.49036
[22] E. Virga:Drops of Nematic liquid crystals, Arch. Rational Mech. Anal.107, (1989), 371-390. · Zbl 0688.76074
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.