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

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.

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 |

### Citations:

Zbl 0262.35020
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\textit{G. H. Williams}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 411--429 (1986; Zbl 0627.49020)

### References:

[1] | Giaquinta, M.; Giusti, E., Global C^{1, α}-regularity for second-order quasilinear elliptic equations in divergence form, J. Reine Angew. Math., t. 351, 55-65, (1984), Preprint · Zbl 0528.35014 |

[2] | Gilbarg, D.; Trudinger, N., Elliptic partial differential equations of second order, (1977), Springer-Verlag Heidelberg, New York · Zbl 0361.35003 |

[3] | Giusti, E., Superfici cartesiane di are minima, Rend. Sem. Mat. Fis. Milano, t. 40, 135-153, (1970) · Zbl 0219.53008 |

[4] | Giusti, E., Boundary behaviour of non-parametric minimal surfaces, Indiana Univ. Math. J., t. 22, 435-444, (1972) · Zbl 0262.35020 |

[5] | Giusti, E., Minimal surfaces and functions of bounded variation, (1984), Birkhaüser-Boston Inc. · Zbl 0545.49018 |

[6] | Jenkins, H.; Serrin, J., The Dirichlet problem for the minimal surface equation in higher dimensions, J. Reine Angew. Math., t. 229, 170-187, (1968) · Zbl 0159.40204 |

[7] | Lieberman, G., The quasilinear Dirichlet problem with decreased regularity at the boundary, Comm. Part. Diff. Equats, t. 6, 437-497, (1981) · Zbl 0458.35039 |

[8] | Lieberman, G., The Dirichlet problem for quasilinear elliptic equations with Hölder continuous boundary values, Arch. Rat. Mech. Anal., t. 79, 305-323, (1982) · Zbl 0497.35010 |

[9] | Liebermann, G., The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data, Comm. Partial Differential Equations, t. 11, 167-229, (1986), Preprint |

[10] | Lorentz, G., Approximation of functions, (1966), Holt, Rinehart and Winston New York · Zbl 0153.38901 |

[11] | Miller, K., Extremal barriers on cones with phragmèn-Lindelöf theorems and other applications, Ann. Mat. Pura Appl., t. 90, 4, 297-329, (1971) · Zbl 0231.35004 |

[12] | Simon, L., Global estimates of Hölder continuity for a class of divergence-form elliptic equations, Arch. Rat. Mech. Anal., t. 56, 253-272, (1974) · Zbl 0295.35027 |

[13] | Simon, L., Boundary behaviour of solutions of the nonparametric least area problem, Bull. Austral. Math. Soc., t. 26, 17-27, (1982) · Zbl 0499.49023 |

[14] | L. Simon, Personal communication. |

[15] | G. Williams, Solutions of the minimal surface equation continuous and discontinuous at the boundary. Preprint. To appear. Comm. Partial Differential Equations. · Zbl 0605.49030 |

[16] | Williams, G., The Dirichlet problem for the minimal surface equation with Lipschitz continuous boundary data, J. Reine Angew. Math., t. 354, 123-140, (1984), To appear · Zbl 0541.35033 |

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