The differential equation Δ𝑥=2𝐻(𝑥ᵤ∧𝑥ᵥ) with vanishing boundary values

Wente, Henry C.

doi:10.1090/S0002-9939-1975-0374673-1

The differential equation $\Delta x=2H(x_{u}\wedge x_{v})$ with vanishing boundary values
HTML articles powered by AMS MathViewer

by Henry C. Wente PDF

Proc. Amer. Math. Soc. 50 (1975), 131-137 Request permission

Abstract:

If $x(u,v)$ is a solution to the system $\Delta x = 2H({x_u} \wedge {x_v})$ on a bounded domain $G \subset {R^2}$ with finite Dirichlet integral and with $x = 0$ on $\partial G$, then $x \equiv 0$ for simply connected $G$, but for doubly-connected $G$ we construct nontrivial solutions.

References

Garrett Birkhoff and Gian-Carlo Rota, Ordinary differential equations, Introductions to Higher Mathematics, Ginn and Company, Boston, Mass.-New York-Toronto, 1962. MR 0138810
R. Courant, Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces, Interscience Publishers, Inc., New York, N.Y., 1950. Appendix by M. Schiffer. MR 0036317
Robert D. Gulliver II, Regularity of minimizing surfaces of prescribed mean curvature, Ann. of Math. (2) 97 (1973), 275–305. MR 317188, DOI 10.2307/1970848
Philip Hartman and Aurel Wintner, On the local behavior of solutions of non-parabolic partial differential equations, Amer. J. Math. 75 (1953), 449–476. MR 58082, DOI 10.2307/2372496
Erhard Heinz, Über die Existenz einer Fläche konstanter mittlerer Krümmung bei vorgegebener Berandung, Math. Ann. 127 (1954), 258–287 (German). MR 70013, DOI 10.1007/BF01361126
Stefan Hildebrandt, Randwertprobleme für Flächen mit vorgeschiebener mittlerer Krümmung und Anwendungen auf die Kapillaritätstheorie. I. Fest vorgebener Rand, Math. Z. 112 (1969), 205–213 (German). MR 250208, DOI 10.1007/BF01110219
Stefan Hildebrandt and Helmut Kaul, Two-dimensional variational problems with obstructions, and Plateau’s problem for $H$-surfaces in a Riemannian manifold, Comm. Pure Appl. Math. 25 (1972), 187–223. MR 296829, DOI 10.1002/cpa.3160250208

Isoperimetrische Ungleichungen und Das Plateausche Probleme

Henry C. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl. 26 (1969), 318–344. MR 243467, DOI 10.1016/0022-247X(69)90156-5
Henry C. Wente, The Dirichlet problem with a volume constraint, Manuscripta Math. 11 (1974), 141–157. MR 328752, DOI 10.1007/BF01184954