Global regularity for solutions of the minimal surface equation with continuous boundary values. (English) Zbl 0627.49020

Williams, Graham H.

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 411-429 (1986).

The author studies the Hölder continuity of solutions of the Dirichlet problem for the minimal surface equation. The first set of results the author obtains generalizes the work of E. Giusti [Indiana Univ. Math. J. 22, 435-444 (1972; Zbl 0262.35020)]: Assuming the boundary data has modulus of continuity \(\beta(t)\) at \(x_ 0\) and that the mean curvature of the boundary grows like \(| x-x_ 0|^{\gamma}\), \(\gamma\geq 0\), then the solution to the Dirichlet problem has modulus of continuity \(C\beta Ct^{1/(2+\gamma)}\) at \(x_ 0\) for some constant C.
In another section, the author defines a function \(K(\alpha)\) on the interval \((0,1)\) such that if the Lipschitz constant of the boundary data (the boundary having nonnegative mean curvature) is less than \(K(\alpha)\), then the solution to the Dirichlet problem is Hölder continuous with exponent \(\alpha\) ; it is also shown that \(K(\alpha)\to 0\) as \(\alpha\to 1\) and \(K(\alpha)\to \infty\) as \(\alpha\to 0.\)
In the final section, the author shows that his function \(K(\alpha)\) is best possible. The proofs involve the construction of barriers.

Reviewer: H.Parks

MSC:

49Q05	Minimal surfaces and optimization
26B35	Special properties of functions of several variables, Hölder conditions, etc.
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

Hölder continuity; Dirichlet problem; minimal surface equation; barriers

Citations:

Zbl 0262.35020

References:

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