×

Existence theorem for a Dirichlet problem with free discontinuity set. (English) Zbl 0713.49003

The following result is proved for a Dirichlet type problem with free discontinuities.
Let \(n\in N\), \(n\geq 2\), let \(\Omega \subset {\mathbb{R}}^ n\) be a bounded domain with \({\mathcal H}^{n-1}(\partial \Omega)<+\infty\); assume that a closed set M exists such that \({\mathcal H}^{n-1}(M)=0\) and \(\partial \Omega \setminus M\) is a \(C^ 1\) surface; let \(w\in C^ 1(\partial \Omega \setminus M)\cap L^{\infty}(\partial \Omega \setminus M)\). Then there exists at least one pair \((K_ 0,u_ 0)\) minimizing the functional \({\mathcal G}\) defined for every closed set \(K\subset {\bar \Omega}\) and for every \(u\in C^ 1(\Omega \setminus K)\cap C^ 0({\bar \Omega}\setminus (M\cup K))\) with \(u=w\) on \(\partial \Omega \setminus (M\cup K)\) by \[ {\mathfrak G}\setminus (K,u)=\int_{\Omega \setminus K}| \nabla u|^ 2dy+{\mathcal H}^{n-1}(K). \] Moreover \(K_ 0\) is (\({\mathcal H}^{n-1},n-1)\) rectifiable, and there exists a unique essential minimizing pair \((K',u')\) (i.e. \(K'\) is the smallest closed set contained in \(K_ 0\) for which \(u_ 0\) has an extension \(u'\in C^ 1(\Omega \setminus K')\cap C^ 0({\bar \Omega}\setminus (M\cup K'))\) with \(u'=w\) on \(\partial \Omega \setminus (M\cup K'))\) such that \[ \liminf_{\rho \to 0}\rho^{1-n}{\mathcal H}^{n-1}(K'\cap \bar B_{\rho}(x))>0 \] for every \(x\in K'\setminus M.\)
Notice that the Dirichlet condition is required on \(\partial \Omega\) but for that part of the boundary that might be touched by the singular set K; however there is a price to be paid for this, because \({\mathcal H}^{n-1}(K)\) penalizes both \(K\cap \Omega\) (where \(u\) is discontinuous) and the part of the boundary \(K\cap \partial \Omega\) where \(u\) may not assume the given value.
A Neumann type problem with free discontinuities has been studied by the authors and E. De Giorgi in Arch. Ration. Mech. Anal. 108, No. 3, 195-218 (1989; Zbl 0682.49002).
Reviewer: M.Carriero

MSC:

49J10 Existence theories for free problems in two or more independent variables

Citations:

Zbl 0682.49002
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Alt, H.W.; Caffarelli, L.A., Existence and regularity for a minimum problem with free boundary, J. reine angew. math., 105, 105-144, (1981) · Zbl 0449.35105
[2] Ambrosio, L., A compactness theorem for a special class of functions of bounded variation, Boll. un. mat. ital., 3-B, 857-881, (1989) · Zbl 0767.49001
[3] Ambrosio L., Existence theory for a new class of variational problems, Archs ration. Mech. Analysis (to appear).
[4] Anzellotti, G.; Giaquinta, M., Funzioni BV e tracce, Rc. semin. mat. univ. Padova, 60, 1-22, (1978) · Zbl 0432.46031
[5] Baiocchi, C.; Capelo, A., Disequazioni variazionali e quasivariazionali, Vols 1 and 2, (1978), Pitagora Bologna · Zbl 1308.49002
[6] Blat, J.; Morel, J.M., Elliptic problems in image segmentation, (1988), preprint · Zbl 0727.35040
[7] Brezis, H.; Coron, J.M.; Lieb, E.H., Harmonic maps with defects, Communs math. phys., 107, 679-705, (1986) · Zbl 0608.58016
[8] Carriero, M.; Leaci, A.; Pallara, D.; Pascali, E., Euler conditions for a minimum problem with free discontinuity surfaces, (1988), Dipartimento di Matematica Lecce, preprint
[9] Chandrasekhar, S., Liquid crystals, (1977), Cambridge University Press Cambridge · Zbl 0127.41403
[10] C{\scongedo} G. & T{\scamanini} I., On the existence of solutions to a problem in multidimensional segmentation, Ann. Inst. H. Poincaré Analyse non Linéaire (to appear).
[11] D{\scal maso} G., M{\scorel} J.M. & S{\scolimini} S., A variational method in image segmentation: existence and approximation results, Acta Math. (to appear).
[12] D{\sce} G{\sciorgi} E., Free discontinuity problems in calculus of variations, Proc. Int. Meeting in J.L. Lion’s honour in J. L. Lion’s honour, Paris, June 6-10, 1988 (to appear).
[13] De, Giorgi E.; Ambrosio, L., Un nuovo tipo di funzionale del calcolo delle variazioni, Atti. accad. naz. lincei, 82, 199-210, (1988)
[14] De, Giorgi E.; Carriero, M.; Leaci, A., Existence theorem for a minimum problem with free discontinuity set, Archs ration. mech. analysis, 108, 195-218, (1989) · Zbl 0682.49002
[15] De, Giorgi E.; Colombini, F.; Piccinini, L.C., Frontiere orientate di misura minima e questioni collegate, () · Zbl 0296.49031
[16] De, Giorgi E.; Congedo, G.; Tamanini, I., Problemi di regolarità per un nuovo tipo di funzionale del calcolo delle variazioni, Atti accad. naz. lincei, 82, (1988)
[17] Ericksen, J.L., Equilibrium theory of liquid crystals, (), 233-299 · Zbl 1171.74019
[18] Federer, H., Geometric measure theory, (1969), Springer Berlin · Zbl 0176.00801
[19] Giusti, E., Minimal surfaces and functions of bounded variation, (1984), Birkhäuser Boston · Zbl 0545.49018
[20] Hardt, R.; Kinderlehrer, D.; Lin, F.H., Existence and partial regularity of static liquid crystal configurations, Communs math. phys., 105, 547-570, (1986) · Zbl 0611.35077
[21] Kinderlehrer, D.; Stampacchia, G., An introduction to variational inequalities and their applications, (1980), Academic Press New York · Zbl 0457.35001
[22] Massari, U.; Miranda, M., Minimal surfaces of codimension one, (1984), North-Holland Amsterdam · Zbl 0565.49030
[23] Morel, J.M.; Solimini, S., Segmentation of images by variational methods: a constructive approach, Rev. mat. univ. complutense Madrid, 1, 169-182, (1988) · Zbl 0679.68205
[24] Mumford, D.; Shah, J., Optimal approximations by piecewise smooth functions and associated variational problems, Communs pure appl. math., 42, 577-685, (1989) · Zbl 0691.49036
[25] Virga, E., Drops of nematic liquid crystals, Archs ration. mech. analysis, 107, 371-390, (1989) · 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.