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Memoirs of the American Mathematical Society

机译：美国数学学会回忆录

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When a domain in the plane is specified by the requirement that thereexists a harmonic function which is zero on its boundary and additionallysatisfies a prescribed Neumann condition there, the boundary is called aBernoulli free boundary. (The boundary is "free" because the domain isnot known a priori and the name Bernoulli was originally associated withsuch problems in hydrodynamics.) Questions of existence, multiplicity oruniqueness, and regularity of free boundaries for prescribed data need to beaddressed and their solutions lead to nonlinear problems. In this paper an equivalence is established between Bernoulli free-boundaryproblems and a class of equations for real-valued functions of one real vari-able. We imposes no restriction on the amplitudes or shapes of free bound-aries, nor on their smoothness. Therefore the equivalence is global, and valideven for very weak solutions. An essential observation here is that the equivalent equations can bewritten as nonlinear Riemann-Hilbert problems and the theory of complexHardy spaces in the unit disc has a central role. An additional useful factis that they have gradient structure, their solutions being critical pointsof a natural Lagrangian. This means that a canonical Morse index canbe assigned to free boundaries and the the Calculus of Variations becomesavailable as a tool for the study. Some rather natural conjectures about the regularity of free boundariesremain unresolved.
机译：当平面中的一个区域由在其边界上存在零的谐波函数并在那里满足规定的Neumann条件的要求指定时，该边界称为伯努利自由边界。 （边界是“自由的”，因为该领域不是先验的，并且名称伯努利最初与流体力学中的此类问题相关。）需要解决规定数据的自由边界的存在性，多重性或唯一性和规则性问题，其解决方案导致非线性问题。在本文中，伯努利自由边界问题和一个实变量的实值函数的方程组之间建立了等价关系。我们没有限制自由边界的幅度或形状，也没有限制它们的平滑度。因此，等价是全局的，即使对于非常弱的解决方案也有效。这里的一个基本观察是，等价方程可以写成非线性Riemann-Hilbert问题，单位圆盘中的复杂Hardy空间理论起着核心作用。另一个有用的事实是它们具有梯度结构，其解是自然拉格朗日方程的临界点。这意味着可以将标准的摩尔斯指数分配给自由边界，并且微积分的微积分可以用作研究的工具。关于自由边界的规律性的一些相当自然的猜想仍然没有解决。

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