[Paper Review] Boundary Expansions for Constant Mean Curvature Surfaces in the Hyperbolic Space

Qing Han, Yue Wang|arXiv (Cornell University)|Aug 28, 2016

Nonlinear Partial Differential Equations18 references3 citations

TL;DR

This paper establishes optimal asymptotic expansions for solutions to the constant mean curvature equation in hyperbolic space near the boundary, using formal expansions involving powers and logarithmic terms of the normal variable. It proves sharp remainder estimates and identifies a slight loss of regularity in the coefficients, even for boundary data of finite regularity, under the condition |H| < 1.

ABSTRACT

We study expansions near the boundary of solutions to the Dirichlet problem for the constant mean curvature equation in the hyperbolic space. With a characterization of remainders of the expansion by multiple integrals, we establish optimal asymptotic expansions of solutions with boundary values of finite regularity and demonstrate a slight loss of regularity for coefficients.

Motivation & Objective

• To analyze the boundary regularity of solutions to the constant mean curvature equation in hyperbolic space for boundary data of finite regularity.
• To derive optimal asymptotic expansions of solutions near the boundary, including logarithmic terms.
• To characterize the remainders in the expansion using multiple integrals and establish sharp estimates.
• To investigate the regularity loss in the coefficients of the expansion, even when the boundary data is smooth.
• To extend results from the minimal surface case (H = 0) to the non-zero mean curvature case (H ≠ 0), highlighting key differences.

Proposed method

• Formal expansion of the solution u in powers and logarithmic powers of the normal variable xₙ, with coefficients cᵢ and cᵢ,ⱼ depending on x′ and H.
• Derivation of explicit expressions for the first few coefficients (c₀, c₁, c₂, ..., cₙ, cₙ₊₁,₁) in terms of φ and H.
• Use of a change of variables to t = xₙ and transformation of the PDE into a form involving ∂ₙₙv, ∂ₙv, and lower-order terms.
• Rewriting the equation in terms of v = u - uₖ to isolate the remainder and analyze its behavior via weighted estimates.
• Application of weighted Sobolev and Schauder estimates to control the remainder u - uₖ in terms of the regularity of φ and H.
• Establishing that the remainder F in the normal derivative equation satisfies a structure allowing for regularity propagation despite logarithmic terms and singular coefficients.

Experimental results

Research questions

• RQ1 How do the asymptotic expansions of solutions to the constant mean curvature equation in hyperbolic space differ from those in the minimal case (H = 0)?
• RQ2 What is the precise structure of the remainder in the expansion of u near the boundary, and how can it be estimated optimally?
• RQ3 To what extent is there a loss of regularity in the coefficients of the expansion, even when the boundary data φ is smooth?
• RQ4 How do logarithmic terms arise in the expansion, and under what conditions are they absent?
• RQ5 Can sharp estimates for the remainder be derived when the boundary data and H have only finite regularity?

Key findings

• The paper establishes optimal asymptotic expansions for solutions to the constant mean curvature equation in hyperbolic space, with remainders estimated via multiple integrals and weighted norms.
• A slight loss of regularity occurs in the coefficients of the expansion, even when the boundary data φ and H are smooth.
• Logarithmic terms appear in the expansion for all dimensions n ≥ 2 when H ≠ 0, except for n = 2 with constant H, where they vanish.
• The first global term in the expansion is cₙ₊₁,₀, which has no explicit expression in terms of φ and H.
• The remainder u - uₖ satisfies a linear PDE with coefficients that are smooth in the data and v, allowing for regularity propagation via weighted estimates.
• The structure of the remainder equation ensures that sharp estimates can be derived even with singular coefficients of order 1/t and logarithmic terms.