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Boundary regularity for minimizing currents with prescribed mean curvature. (English) Zbl 0806.49028

The authors prove complete boundary regularity for energy minimizing integer multiplicity rectifiable \(n\) currents \(T\) in \(\mathbb{R}^{n+1}\) of prescribed mean curvature \(H\) with boundary \(B= \partial T\) represented by an oriented smooth submanifold of dimension \(n-1\) in \(\mathbb{R}^{n+1}\). Also, combining their boundary regularity theorem with the interior regularity theory, they formulate and prove a theorem regarding the Plateau problem for surfaces with prescribed mean curvature.
The notion of energy minimality here refers to the energy functional \(E_ H\) associated with a prescribed mean curvature function \(H: \mathbb{R}^{n+1}\to \mathbb{R}\), i.e., \(E_ H(T)= M(T)- V_ H(T)\), where \(M(T)\) is the mass (= total \(n\) area) of \(T\) and \(V_ H(T)\) is the \(H\) weighted volume enclosed by \(T\) and a fixed integer multiplicity rectifiable reference \(n\) current \(T_ 0\) with \(\partial T_ 0= B\).
The references contain 35 papers, but the indispensable work for the preceding results is R. Hardt and L. Simon [Ann. Math., II. Ser. 110, 439-486 (1979; Zbl 0457.49029)].

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
49Q05 Minimal surfaces and optimization

Citations:

Zbl 0457.49029
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References:

[1] Allard, W.K.: On boundary regularity for Plateaus’s problem. Bull. Am. Math. Soc.75, 522-523 (1969) · Zbl 0183.11404 · doi:10.1090/S0002-9904-1969-12229-9
[2] Allard, W.K.: On the first variation of a varifold. Ann. Math.95, 417-491 (1972) · Zbl 0252.49028 · doi:10.2307/1970868
[3] Allard, W.K.: On the first variation of a varifold: boundary behaviour. Ann. Math.101, 418-446 (1975) · Zbl 0319.49026 · doi:10.2307/1970934
[4] Almgren, Jr., F.J.: Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure. Ann. Math.87, 321-391 (1968) · Zbl 0162.24703 · doi:10.2307/1970587
[5] Alt, H.W.: Verzweigungspunkte vonH-Flächen. I, II. Math. Z.127, 333-362 (1972); Math. Ann.201, 33-55 (1973) · Zbl 0253.58007 · doi:10.1007/BF01111392
[6] Brothers, J.: Existence and structure of tangent cones at the boundary of an area minimizing integral current. J. Indian. Univ. Math.26, 1027-1044 (1977) · Zbl 0374.49022 · doi:10.1512/iumj.1977.26.26083
[7] Barbosa, J.L., Do Carmo, M.: On the size of a stable minimal surface in ?3. Am. J. Math.98, 515-528 (1976) · Zbl 0332.53006 · doi:10.2307/2373899
[8] Berger, M., Gauduchon, P., Mazet, E.: Le spectre d’une variété Riemannienne. (Lect. Notes Math., vol. 194) Berlin, Heidelberg, New York: Springer 1971 · Zbl 0223.53034
[9] Duzaar, F.: On the existence of surfaces with prescribed mean curvature and boundary in higher dimensions. Analyse Non linéaire10, 191-214 (1992) · Zbl 0808.49036
[10] Duzaar, F.: Boundary regularity for area minimizing currents with prescribed volume. Preprint no. 265, SFB 256 Bonn 1992. · Zbl 0787.53056
[11] Duzaar, F., Fuchs, M.: On the existence of integral currents with prescribed mean curvature vector. Manuscr. Math.67, 41-67 (1990) · Zbl 0703.49035 · doi:10.1007/BF02568422
[12] Duzaar, F., Fuchs, M.: A general existence theorem for integral currents with prescribed mean curvature form. Boll. U. M. I. (7)6, 901-912 (1992) · Zbl 0786.49025
[13] De Giorgi, E.: Frontiere orientate di misura minima. Semin. Mat. Sc. Norm. Sup. Pisa, 1-56 (1961)
[14] De Giorgi, E., Colombini, F., Piccinini, L.C.: Frontiere orientate di misura minima e questioni collegate. Scuola Norm. Sup. Pisa 1972 · Zbl 0296.49031
[15] Duzaar, F., Steffen, K.: Area minimizing hypersurfaces with prescribed volume and boundary. Math. Z.209, 581-618 (1992) · Zbl 0787.53056 · doi:10.1007/BF02570855
[16] Duzaar, F., Steffen, K.: Comparison principles for hypersurfaces of prescribed mean curvature. Preprint 1993 · Zbl 0806.49028
[17] Duzaar, F., Steffen, K.: ? minimizing currents. To appear in Manuscr. Math. · Zbl 0819.53034
[18] Federer, H.: Geometric measure theory. Berlin, Heidelberg, New York: Springer 1969 · Zbl 0176.00801
[19] Federer, H.: The singular set of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension. Bull. Am. Math. Soc.76, 767-771 (1970) · Zbl 0194.35803 · doi:10.1090/S0002-9904-1970-12542-3
[20] Gulliver, R.: Regularity of minimizing surfaces of prescribed mean curvature. Ann. Math.97, 275-305 (1973) · Zbl 0246.53053 · doi:10.2307/1970848
[21] Gulliver, R., Lesley, F.D.: On boundary branch points of minimizing surfaces. Arch. Ration. Mech. Anal.52, 20-25 (1973) · Zbl 0263.53009 · doi:10.1007/BF00249089
[22] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. 2nd. edn. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0361.35003
[23] Hardt, R.: On boundary regularity for integral currents or flat chains modulo two minimizing the integral of an elliptic integrand. Commun. Partial Differ. Equations2, 1163-1232 (1977) · Zbl 0385.49025 · doi:10.1080/03605307708820058
[24] Heinz, E.: Über das Randverhalten quasilinearer elliptischer Systeme mit isothermen Parametern. Math. Z.113, 99-105 (1970) · Zbl 0181.11404 · doi:10.1007/BF01141095
[25] Hildebrandt, S.: Randwertprobleme für Flächen mit vorgeschriebener mittlerer Krümmung und Anwendungen auf die Kapillaritätstheorie I. Math. Z.112, 205-213 (1969) · Zbl 0177.15104 · doi:10.1007/BF01110219
[26] Hardt, R., Lin, F.H.: Tangential regularity near theC 1-boundary. Am. Math. Soc. Proc. Symp. Pure Math.44, 245-253 (1986)
[27] Hardt, R., Simon, L.: Boundary regularity and embedded solutions for the oriented Plateau problem. Ann. Math.110, 439-486 (1979) · Zbl 0457.49029 · doi:10.2307/1971233
[28] Ladyshenskaya, O.A., Ural’tseva, N.N.: Linear and quasilinear elliptic equations. New York, London: Academic Press 1968
[29] Massari, U.: Esistenza e regolarità delle ipersuperfici di curvatura media assegnata in ? n . Arch. Ration. Mech. Analysis55, 357-382 (1974) · Zbl 0305.49047 · doi:10.1007/BF00250439
[30] Morrey, C.B.: Multiple integrals in the calculus of variations. Berlin, Heidelberg, New York: Springer 1966 · Zbl 0142.38701
[31] Massari, U., Miranda, M.: Minimal surfaces of codimension one. (Mathematics Studies, vol. 91), Amsterdam: North-Holland 1984 · Zbl 0565.49030
[32] Simon, L.: Lectures on geometric measure theory. Proc. Cent. Math. Anal. Vol.3, Canberra: Austr. Nat. Univ. 1983 · Zbl 0546.49019
[33] Steffen, K.: On the existence of surfaces with prescribed mean curvature and boundary. Math. Z.146, 113-135 (1976) · Zbl 0343.49016 · doi:10.1007/BF01187700
[34] Taylor, J.E.: The structure of singularities in solutions to ellipsoidal variational problems with constaints in ?3. Ann. Math.103, 541-546 (1976) · Zbl 0335.49033 · doi:10.2307/1970950
[35] White, B.: Regularity of area-minimizing hypersurfaces at boundaries with multiplicities. Seminar on Minimal Submanifolds, 293-301. Princeton: Princeton University Press 1983 · Zbl 0528.53051
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