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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
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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
[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: 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: Dipartimento di Matematica Lecce, preprint
[9] Chandrasekhar, S., Liquid Crystals (1977), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0127.41403
[13] De, Giorgi E.; Ambrosio, L., Un nuovo tipo di funzionale del calcolo delle variazioni, Atti. Accad. naz. Lincei, 82, 199-210 (1988) · Zbl 0715.49014
[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, (Quaderni S.N.S. (1972), Editrice Tecnico Scientifica: Editrice Tecnico Scientifica Pisa) · 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, (Brown, G. H., Adv. Liq. Cryst., Vol. 2 (1976), Academic Press: Academic Press New York), 233-299 · Zbl 1171.74019
[18] Federer, H., Geometric Measure Theory (1969), Springer: Springer Berlin · Zbl 0176.00801
[19] Giusti, E., Minimal Surfaces and Functions of Bounded Variation (1984), Birkhäuser: 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: Academic Press New York · Zbl 0457.35001
[22] Massari, U.; Miranda, M., Minimal Surfaces of Codimension One (1984), North-Holland: 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
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