Sharp size estimates for capillary free surfaces without gravity

Ma, Xiang

doi:10.2140/pjm.2000.192.121

Cited by 9 publications

(4 citation statements)

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“…For completeness, we will prove Theorem (5.2) to end this paper. Our proof is similar to that in [22] except for the different signs in somewhere.…”

Section: Sosupporting

confidence: 75%

“…What if the second alternative happens? We answer this in the following theorem whose Euclidean version was proved by Xi-Nan Ma in [22].…”

Section: Nonuniqueness Of the Minimal Pointmentioning

confidence: 96%

“…But he did not assert the sharpness of the estimates, since he did not have a similar minimum principle for Φ(x, 1) at that time. In 2000, Xi-Nan Ma [22] solved this issue through uniqueness of critical point and analyticity of the solution. He took a long computation to show that all the derivatives of Φ(x, 1) vanish at the unique critical point if Φ(x, 1) takes its minimum value at that point.…”

Section: Introdutionmentioning

confidence: 99%

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A sharp height estimate for the spacelike constant mean curvature graph in the Lorentz–Minkowski space

Zhu

2017

Pacific J. Math.

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Abstract. In this paper, based on the local comparison principle in [12], we study the local behavior of the difference of two spacelike graphs in a neighborhood of a second contact point. Then we apply it to the constant mean curvature equation in 3-dimensional LorentzMinkowski space L 3 and get the uniqueness of critical point for the solution of such equation over convex domain, which is an analogue of the result in [28]. Last, by this uniqueness, we obtain a minimum principle for a functional depending on the solution and its gradient. This gives us a sharp gradient estimate for the solution, which leads to a sharp height estimate. IntrodutionSpacelike constant mean curvature(CMC) hypersurfaces and CMC foliation play an important role in general relativity. Such surfaces are important because they provide Riemannian submanifolds with properties which reflect those of the spacetime. For example, if the weak energy condition is satisfied, then a maximal hypersurface has positive scalar curvature. So the geometric properties of such hypersurfaces are worth researching. In particular, the existence of such hypersurface is a fundamental problem. Under the graph setting and some assumptions, Robert Bartnik and Leon Simon[4] got a sufficient and necessary condition for the existence ofwhere div stands for divergence operator in the Euclidean plane R n andIn particular, the Theorem 3.6 in [4] gives us a solution u ∈ C ∞ (Ω) toover a bounded C 2,α domain Ω with H being a positive constant. In this case, they pointed out that ν n+1 =

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“…For completeness, we will prove Theorem (5.2) to end this paper. Our proof is similar to that in [22] except for the different signs in somewhere.…”

Section: Sosupporting

confidence: 75%

“…What if the second alternative happens? We answer this in the following theorem whose Euclidean version was proved by Xi-Nan Ma in [22].…”

Section: Nonuniqueness Of the Minimal Pointmentioning

confidence: 96%

Section: Introdutionmentioning

confidence: 99%

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A sharp height estimate for the spacelike constant mean curvature graph in the Lorentz–Minkowski space

Zhu

2017

Pacific J. Math.

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“…Various special cases of (1.1) have been considered in [4,5,7,8,10,11,13,14,16]. In this paper, we will consider another special case of the form (g(u)u, i ), i +f (x, u, q) = 0 in D (1.2) or its differentiated form…”

Section: Introductionmentioning

confidence: 99%