Abstract
In this work we study solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. With such a solution we can naturally associate a current with support in the closed cylinder above the domain and with boundary given by the prescribed boundary data and which inherits a natural minimizing property. Our main result is that its support is a C 1,α manifold-with-boundary, with boundary equal to the prescribed boundary data, provided that both the initial domain and the prescribed boundary data are of class C 1,α.
Similar content being viewed by others
References
Allard, W.K.: On the first variation of a varifold. Ann. Math. 2 95, 417–491 (1972). MR MR0307015 (46 #6136)
Allard, W.K.: On the first variation of a varifold: boundary behavior. Ann. Math. 2 101, 418–446 (1975). MR MR0397520 (53 #1379)
Bombieri, E., Giusti, E.: Local estimates for the gradient of non-parametric surfaces of prescribed mean curvature. Commun. Pure Appl. Math. 26, 381–394 (1973). MR MR0344977 (49 #9716)
Brézis, H., Kinderlehrer, D.: The smoothness of solutions to nonlinear variational inequalities. Indiana Univ. Math. J. 23, 831–844 (1973/1974). MR MR0361436 (50 #13881)
Barozzi, E., Massari, U.: Regularity of minimal boundaries with obstacles. Rend. Semin. Mat. Univ. Padova 66, 129–135 (1982). MR MR664576 (83m:49066)
Caffarelli, L.A.: Compactness methods in free boundary problems. Commun. Partial Differ. Equ. 5(4), 427–448 (1980). MR MR567780 (81e:35121)
Duzaar, F., Steffen, K.: Boundary regularity for minimizing currents with prescribed mean curvature. Calc. Var. Partial Differ. Equ. 1(4), 355–406 (1993). MR MR1383909 (97e:49036)
Duzaar, F., Steffen, K.: λ minimizing currents. Manuscr. Math. 80(4), 403–447 (1993). MR MR1243155 (95f:49062)
Finn, R.: Remarks relevant to minimal surfaces, and to surfaces of prescribed mean curvature. J. Anal. Math. 14, 139–160 (1965). MR MR0188909 (32 #6337)
Gerhardt, C.: Global C 1,1-regularity for solutions of quasilinear variational inequalities. Arch. Ration. Mech. Anal. 89(1), 83–92 (1985). MR MR784104 (87f:49013)
Giusti, E.: Superfici cartesiane di area minima. Rend. Semin. Mat. Fis. Milano 40, 135–153 (1970). MR MR0291963 (45 #1051)
Giusti, E.: Boundary behavior of non-parametric minimal surfaces. Indiana Univ. Math. J. 22, 435–444 (1972/1973). MR MR0305253 (46 #4383)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 edn. MR MR1814364 (2001k:35004)
Hardt, R., Simon, L.: Boundary regularity and embedded solutions for the oriented Plateau problem. Bull. Am. Math. Soc. (N.S.) 1(1), 263–265 (1979). MR MR513755 (81b:53012)
Jenkins, H., Serrin, J.: The Dirichlet problem for the minimal surface equation in higher dimensions. J. Reine Angew. Math. 229, 170–187 (1968). MR MR0222467 (36 #5519)
Korevaar, N.J., Simon, L.: Continuity estimates for solutions to the prescribed-curvature Dirichlet problem. Math. Z. 197(4), 457–464 (1988). MR MR932680 (89f:35082)
Lin, F.-H.: Behaviour of nonparametric solutions and free boundary regularity. In: Miniconference on Geometry and Partial Differential Equations, 2, Canberra, 1986. Proceedings of the centre fir mathematical analysis. Australian National University, vol. 12, pp. 96–116. Australian National University, Canberra (1987). MR MR924431 (89c:35047)
Lau, C.P., Lin, F.-H.: The best Hölder exponent for solutions of the nonparametric least area problem. Indiana Univ. Math. J. 34(4), 809–813 (1985). MR MR808827 (87d:49058)
Ladyzhenskaya, O.A., Ural’tseva, N.N.: Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations. Commun. Pure Appl. Math. 23, 677–703 (1970). MR MR0265745 (42 #654)
Miranda, M.: Frontiere minimali con ostacoli. Ann. Univ. Ferrara Sez. VII (N.S.) 16, 29–37 (1971). MR MR0301617 (46 #773)
Miranda, M.: Un principio di massimo forte per le frontiere minimali e una sua applicazione alla risoluzione del problema al contorno per l’equazione delle superfici di area minima. Rend. Semin. Mat. Univ. Padova 45, 355–366 (1971). MR MR0303390 (46 #2527)
Miranda, M.: Dirichlet problem with L 1 data for the non-homogeneous minimal surface equation. Indiana Univ. Math. J. 24, 227–241 (1974/1975). MR MR0352682 (50 #5169)
Serrin, J.: The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables. Philos. Trans. R. Soc. Lond. Ser. A 264, 413–496 (1969). MR MR0282058 (43 #7772)
Simon, L.: Global estimates of Hölder continuity for a class of divergence-form elliptic equations. Arch. Ration. Mech. Anal. 56, 253–272 (1974). MR MR0352696 (50 #5183)
Simon, L.: Boundary regularity for solutions of the non-parametric least area problem. Ann. Math. 2 103(3), 429–455 (1976). MR MR0638358 (58 #30681)
Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Centre for Mathematical Analysis. Australian National University, vol. 3. Australian National University Centre for Mathematical Analysis, Canberra (1983). MR MR756417 (87a:49001)
Tamanini, I.: Boundaries of Caccioppoli sets with Hölder-continuous normal vector. J. Reine Angew. Math. 334, 27–39 (1982). MR MR667448 (83m:49067)
Trudinger, N.S.: Gradient estimates and mean curvature. Math. Z. 131, 165–175 (1973). MR MR0324187 (48 #2539)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by John M. Lee.
Rights and permissions
About this article
Cite this article
Bourni, T. C1,α Theory for the Prescribed Mean Curvature Equation with Dirichlet Data. J Geom Anal 21, 982–1035 (2011). https://doi.org/10.1007/s12220-010-9176-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-010-9176-6