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Global regularity for solutions of the minimal surface equation with continuous boundary values

Journal Article


Abstract


  • Suppose Ω is a bounded open subset of ℝn with C2 boundary ∂Ω having nonnegative mean curvature. We examine the regularity at the boundary of solutions u to the minimal surface equation having boundary values ϕ. If ϕ has modulus of continuity β we give a modulus of continuity for u which depends on β and the behaviour of the mean curvature of ∂Ω. If ϕ is Lipschitz continuous then we show that u is Hölder continuous with some exponent α (explicitly obtained) that depends on the Lipschitz constant for ϕ. Finally we give examples showing the above results are best possible.

Publication Date


  • 1986

Citation


  • Williams, G. H. (1986). Global regularity for solutions of the minimal surface equation with continuous boundary values. Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, 3(6), 411-429. doi:10.1016/S0294-1449(16)30375-4

Scopus Eid


  • 2-s2.0-84990281078

Web Of Science Accession Number


Start Page


  • 411

End Page


  • 429

Volume


  • 3

Issue


  • 6

Abstract


  • Suppose Ω is a bounded open subset of ℝn with C2 boundary ∂Ω having nonnegative mean curvature. We examine the regularity at the boundary of solutions u to the minimal surface equation having boundary values ϕ. If ϕ has modulus of continuity β we give a modulus of continuity for u which depends on β and the behaviour of the mean curvature of ∂Ω. If ϕ is Lipschitz continuous then we show that u is Hölder continuous with some exponent α (explicitly obtained) that depends on the Lipschitz constant for ϕ. Finally we give examples showing the above results are best possible.

Publication Date


  • 1986

Citation


  • Williams, G. H. (1986). Global regularity for solutions of the minimal surface equation with continuous boundary values. Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, 3(6), 411-429. doi:10.1016/S0294-1449(16)30375-4

Scopus Eid


  • 2-s2.0-84990281078

Web Of Science Accession Number


Start Page


  • 411

End Page


  • 429

Volume


  • 3

Issue


  • 6