Free boundary minimal surfaces and overdetermined boundary value problems
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NIKOLAI S. NADIRASHVILI AND ALEXEI V. PENSKOI∗ Dedicated to Lawrence Zalcman Abstract. In this paper we establish a connection between free boundary minimal surfaces in a ball in R3 and free boundary cones arising in a one-phase problem.
1
Introduction
Consider free boundary minimal surfaces in a three-dimensional ball, i.e., proper branched minimal immersions of a surface M u : M −→ B3 ⊂ R3 such that u(M) meets the boundary sphere S2 = ∂B3 orthogonally. It is a classical and developed subject; see, e.g., the books [DHS10, DHT10a, DHT10b]. A celebrated result due to J. C. C. Nitsche [Nit72] states that if M is a disk and u is an embedding, then u(M) is also a plane disk. Actually, in the paper [Nit72] a stronger result is announced, namely, that this statement holds for capillary surfaces and the angle between u(M) and S2 is a constant. Details of the proof could be found in the paper [RS97]. Recently, the result due to J. C. C. Nitsche was generalized by A. Fraser and R. Schoen in the paper [FS15] to the case of an immersed minimal disk satisfying the free boundary condition in a constant curvature ball of any dimension. ∗ The work of the second author was partially supported by the Simons Foundation and by the Young Russian Mathematics award.
323 ´ JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 141 (2020) DOI 10.1007/s11854-020-0129-0
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N. S. NADIRASHVILI AND A. V. PENSKOI
A. Fraser and R. Schoen also proved the existence of free boundary minimal surfaces in B3 which have genus 0 and n boundary components, see the papers [FS16], see also [FPZ17] . Let us remark that A. Fraser and R. Schoen established in the papers [FS11, FS16] a remarkable connection between minimal surfaces with free boundaries in a ball and Riemannian metrics on surfaces with boundaries extremizing eigenvalues of the Steklov problem on these surfaces. Let us also remark that a connection of spectral isoperimetry with minimal surfaces was first established by the first author in the paper [Nad96]. In this paper we establish, by means of Proposition 1.2 due to Minkowski, a connection between free boundary minimal surfaces and the extremal domains on the sphere S2 for the Dirichlet problem. The last spectral problem is related to the one-phase free boundary problem for homogeneous functions defined in cones. By virtue of this connection we prove some new results for the one-phase free boundary problem. Minimizers of the one-phase energy functional (|∇v |2 + 1)dx → min, J(v ) = G∩{v>0}
where v 0, are weak solutions of the one-phase free boundary problem ⎧ ⎪ in G, ⎪ ⎨v 0 (1) v = 0 on G+ , ⎪ ⎪ ⎩ |∇v | = 1 on ∂G+ ∩ G, where G+ = {x ∈ G, v (x) > 0}. The gradient condition |∇v | = 1 is satisfied in a viscosity sense. The onephase free boundary problem arises in models of cavitational flow; see, e.g., the book [Fri88] . H. W. Alt and L. A. Caffarelli proved in the paper [AC81] that the question of regularity of the boundary in the one-phase free boundary problem could be reduced to the one-phase problem in a cone. Let K ⊂ Rn be an open (n-dimens
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