Abstract
In this article we consider the Dirichlet problem for hypersurfaces of aniso- tropic prescribed mean curvature H = H(x, u, N) depending on \({x \in \varOmega \subset \mathbb {R}^n}\), the height u of the hypersurface M = graph u over \({\varOmega}\) and the unit normal N to M at (x, u). We give a condition relating H and the mean curvature of \({\partial \varOmega}\) that guarantees the existence of smooth solutions even for not necessarily convex domains.
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References
Bergner M.: The Dirichlet problem for graphs of prescribed anisotropic mean curvature in \({\mathbb {R}^{n+1} }\) . Int. Math. J. Anal. Appl. 28, 149–166 (2008)
Giaquinta M.: On the Dirichlet problem for surfaces of prescribed mean curvature. Manus. Math. 12, 73–86 (1974)
Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)
Serrin, J.: Gradient estimates for solutions of nonlinear elliptic and parabolic equations. Contributions to Nonlinear Functional Analysis, pp. 565–601. Academic Press, New York (1971)
Stampacchia G.: Equations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier 15, 189–258 (1965)
Wang X.-J.: Interior gradient estimates for mean curvature equations. Math. Z. 228, 73–81 (1998)
Acknowledgments
The author wants to thank Prof.Dr. Gerhard Huisken for the fruitful conversations which lead to this result.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Marquardt, T. Remark on the anisotropic prescribed mean curvature equation on arbitrary domains. Math. Z. 264, 507–511 (2010). https://doi.org/10.1007/s00209-009-0476-0
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DOI: https://doi.org/10.1007/s00209-009-0476-0