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On the Dirichlet problem of prescribed mean curvature equations without H-convexity condition

Published online by Cambridge University Press:  22 January 2016

Kazuya Hayasida  and
Masao Nakatani
Kazuya Hayasida
Affiliation:
Department of Computational Science, Faculty of Science, Kanazawa University, Kanazawa, 920-1192, Japan
Masao Nakatani
Affiliation:
Department of Computational Science, Faculty of Science, Kanazawa University, Kanazawa, 920-1192, Japan
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Abstract

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The Dirichlet problem of prescribed mean curvature equations is well posed, if the boundery is H-convex. In this article we eliminate the H-convexity condition from a portion Γ of the boundary and prove the existence theorem, where the boundary condition is satisfied on Γ in the weak sense.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 157 , 2000 , pp. 177 - 209
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2000

References

[1] Gerhardt, C., Existence, regularity, and boundary behaviour of generalized surfaces of prescribed mean curvature, Math. Z., 139 (1974), 173–198.Google Scholar
[2] Giaquinta, M., On the Dirichlet problem for surfaces of prescribed mean curvature, Manuscripta Math., 12 (1974), 73–86.Google Scholar
[3] Gilberg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Springer, New York, 1983.Google Scholar
[4] Giusti, A., Boundary value problems for non-parametric surfaces of prescribed mean curvature, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3 (1976), 501–548.Google Scholar
[5] Gregori, G., Generalized solutions for a class of non-uniformly elliptic equations in divergence form, Comm. Partial Differential Equations, 22 (1997), 581–617.Google Scholar
[6] Ivochkina, N.M., The Dirichlet problem for the equations of curvature of order m, Leningrad Math. J., 2 (1991), 631–654.Google Scholar
[7] Jenkins, H. and Serrin, J., The Dirichlet problem for the minimal surface equation in higher dimensions, J. Reine Angew. Math., 229 (1968), 170–187.Google Scholar
[8] Jin, Z. and Lancaster, K., Behavior of solutions for some Dirichlet problems near reentrant corners, Indiana Univ. Math. J., 46 (1997), 827–862.Google Scholar
[9] Korevaar, N.J. and Simon, L., Continuity estimates for solutions to the prescribed-curvature Dirichlet problem, Math. Z., 197 (1987), 457–464.Google Scholar
[10] Ladyzhenskaya, O.A. and Ural’ceva, N.N., Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations, Comm. Pure Appl. Math., 23 (1970), 677–703.Google Scholar
[11] Ladyzhenskaya, O.A. and Ural’ceva, N.N., Local estimates of the gradients of solution to a simplest regularization for some class of non uniformly elliptic equations, Zap. Nauchn. Sem. S.-Peterburg Otdel Mat. Inst. Steklov, 213 (1994), 75–92, (Russian).Google Scholar
[12] Lancaster, K.E., Nonexistence of some nonparametric surfaces of prescribed mean curvature, Proc. Amer. Math. Soc, 96 (1986), 187–188.Google Scholar
[13] Lau, C.P. and Lin, F. H., The best Hölder exponent for solutions of the non-parametric least area problem, Indiana Univ. Math. J., 34 (1985), 809–813.Google Scholar
[14] Lau, C.P., Quasilinear elliptic equations with small boundary data, Manuscripta Math., 53 (1985), 77–99.Google Scholar
[15] Lichnewsky, A., Sur le comportement au bord des solutions généralisées du problème non paramétrique des surfaces minimals, J. Math. Pures Appl., 53 (1974), 397–476.Google Scholar
[16] Schultz, F. and Williams, G. H., Barriers and existence results for a class of equations of mean curvature type, Analysis, 7 (1987), 359–374.Google Scholar
[17] Serrin, J., The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Phil. Trans. Royal Soc. London, A 264 (1969), 413–496.Google Scholar
[18] Simon, L., Interior gradient bounds for non-uniformly elliptic equations, Indiana Univ. Math. J., 25 (1976), 821–855.Google Scholar
[19] Simon, L., Boundary behaviour of solutions of the non-parametric least area problem, Bull. Austral. Math. Soc, 26 (1982), 17–27.Google Scholar
[20] Teman, R., Applications de l’analyse convexe au calcul des variations, Lecture Notes in Math., 543, Springer, 1975.Google Scholar
[21] Tersenov, A.S., On quasilinear non-uniformly elliptic equations in some non-convex domains, Comm. Partial Differential Equations, 23 (1998), 2165–2185.Google Scholar
[22] Williams, G.H., The Dirichlet problem for the minimal surface equation with Lipschitz continuous boundary data, J. Reine Anegew. Math., 354 (1984), 123–140.Google Scholar
[23] Williams, G.H., Solutions of the minimal surface equation-continuous and discontinuous at the boundary, Comm. Partial Differential Equations, 11 (1986), 1439–1457.Google Scholar