# Bernoulli Free-Boundary Problems

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*E. Shargorodsky; J. F. Toland*

When a domain in the plane is specified by the
requirement that there exists a harmonic function which is zero on its
boundary and additionally satisfies a prescribed Neumann condition
there, the boundary is called a Bernoulli free boundary. (The boundary
is “free” because the domain is not known a
priori and the name Bernoulli was originally associated with such
problems in hydrodynamics.) Questions of existence, multiplicity or
uniqueness, and regularity of free boundaries for prescribed data need
to be addressed and their solutions lead to nonlinear problems.

In this paper an equivalence is established between Bernoulli
free-boundary problems and a class of equations for real-valued
functions of one real variable. The authors impose no restriction on
the amplitudes or shapes of free boundaries, nor on their
smoothness. Therefore the equivalence is global, and valid even for
very weak solutions.

An essential observation here is that the equivalent equations can
be written as nonlinear Riemann-Hilbert problems and the theory of
complex Hardy spaces in the unit disc has a central role. An
additional useful fact is that they have gradient structure, their
solutions being critical points of a natural Lagrangian. This means
that a canonical Morse index can be assigned to free boundaries and
the Calculus of Variations becomes available as a tool for the
study.

Some rather natural conjectures about the regularity of free
boundaries remain unresolved.

#### Table of Contents

# Table of Contents

## Bernoulli Free-Boundary Problems

- Contents v6 free
- Chapter 1. Introduction 110 free
- Chapter 2. Bernoulli Free Boundaries 312 free
- 2.1. Special case: steady hydrodynamic waves 413
- 2.2. General Case 413
- 2.3. Notation 716
- 2.4. Formulation as a Single Equation 918
- 2.5. Equations 918
- 2.6. Example of (2.7) with Explicit Solutions 1019
- 2.7. Equivalence 1120
- 2.8. Inequalities 1221
- 2.9. Duality 1322
- 2.10. Example of (2.7) with Explicit Solutions: Duality 1322
- 2.11. Self-duality 1423

- Chapter 3. Type-(I) Problems 1524
- 3.1. Regularity 1524
- 3.2. Example of (2.7) with Explicit Solutions: Regularity 1625
- 3.3. Dimension of the Set of Stagnation Points 1625
- 3.4. Jordan Curves 1625
- 3.5. Example of (2.7) with Explicit Solutions: Jordan Curves 1726
- 3.6. Nekrasov's Equation 1726
- 3.7. Nekrasov Duality 1928
- 3.8. Example of (2.7) with Explicit Solutions: Nekrasov Duality 2029
- 3.9. Morse Index of Non-singular Solutions 2029
- 3.10. Example of (2.7) with Explicit Solutions: Morse Index 2332
- 3.11. Stokes Waves 2332

- Chapter 4. Proofs of Main Results 2736
- 4.1. Equations: proofs of Theorem 2.4 and Corollary 2.5 2736
- 4.2. Equivalence: proofs of Theorems 2.7, 2.8 and 2.9 2837
- 4.3. Inequalities: proof of Theorem 2.10 3443
- 4.4. Duality 3544
- 4.5. Regularity: proofs of Theorems 3.1 and 3.3 3847
- 4.6. Dimension of the Set of Stagnation Points: proof of Theorem 3.4 4150
- 4.7. Jordan Curves: proofs of Theorem 3.5 and (3.3) 4251
- 4.8. Nekrasov's Equation: proof of Theorem 3.8 4857
- 4.9. Morse Indices 4958
- 4.10. Plotnikov's Transformation 5059
- 4.11. Sign of the Plotnikov Potential 5362
- 4.12. Constant Plotnikov Potentials: Proofs of Theorem 3.14 5362
- 4.13. Simple Morse-Index Estimates: Proof of Lemma 3.10 5463
- 4.14. Morse Index and Stagnation Points 5463
- 4.15. Proof of Theorem 3.13 6170

- Appendix A. Auxiliary results 6372
- Bibliography 6574
- Index 6978 free