Global boundedness, interior gradient estimates, and boundaryregularity for the mean curvature equation with boundaryconditions

Liang, Fei-Tsen

doi:10.1155/s0161171204307039

Cited by 2 publications

(2 citation statements)

References 17 publications

(12 reference statements)

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“…In contrast, the following estimates for the boundary oscillation of u are established in [8,Main Theorem III]. Theorem 1.…”

Section: I(v) +mentioning

confidence: 99%

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Boundary Regularity for Capillary Surfaces

Liang

2005

GMJ

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For solutions of capillarity problems with the boundary contact angle being bounded away from 0 and π and the mean curvature being bounded from above and below, we show the Lipschitz continuity of a solution up to the boundary locally in any neighborhood in which the solution is bounded and ∂Ω is 𝐶2; the Lipschitz norm is determined completely by the upper bound of | cos θ|, together with the lower and upper bounds of 𝐻, the upper bound of the absolute value of the principal curvatures of ∂Ω and the dimension 𝑛.

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“…In contrast, the following estimates for the boundary oscillation of u are established in [8,Main Theorem III]. Theorem 1.…”

Section: I(v) +mentioning

confidence: 99%

“…Assume that there exists a nonnegative constant H * such that |H(x, t)| ≤ H * for x ∈ Ω and t ∈ R. Consider the identity (0.7). Assuming |u| is bounded up to the boundary, [8,Theorem 1] assures us of that u ∈ H 1,1 (Ω) and thus we are allowed to set in (0.7)…”

Section: And Ifmentioning

confidence: 99%

Boundary Regularity for Capillary Surfaces

Liang

2005

GMJ

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Boundary behavior of capillary surfaces possibly with extremal boundary angles

Liang

2005

International Journal of Mathematics and Mathematical Sciences

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For solutions to the capillarity problem possibly with the boundary contact angle θ being 0 and/or π in a relatively open portion of the boundary which is C 2 , we will show that if the solution is locally bounded up to this portion of boundary, the trace of the solution on this portion is piecewise Lipschitz continuous and the solution is Hölder continuous up to the boundary, provided the prescribed mean curvature is bounded from above and from below. In the case where θ is not required to be bounded away from π/2, 0, and π, and the mean curvature H(x,t 0 ) belongs to L p (Ω) for some t 0 ∈ R and p > n, under the assumption that in a neighborhood of a relatively open portion of the boundary the solution is of rotational symmetry, the trace of the solution on this portion of the boundary is shown to be Hölder continuous with exponent 1/n if n ≥ 3 and with exponent 1/3 if n = 2.

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Global boundedness, interior gradient estimates, and boundaryregularity for the mean curvature equation with boundaryconditions

Cited by 2 publications

References 17 publications

Boundary Regularity for Capillary Surfaces

Boundary Regularity for Capillary Surfaces

Boundary behavior of capillary surfaces possibly with extremal boundary angles

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