# A free boundary problem for the Laplacian with a constant Bernoulli-type boundary condition

Abstract: We study a free boundary problem for the Laplace operator, where we impose a Bernoulli-type boundary condition. We show that there exists a solution to this problem. We use A. Beurling's technique, by defining two classes of sub- and supersolutions and a Perron argument. We try to generalize here a previous work of A. Henrot and H. Shahgholian. We extend these results in different directions.

## Summary (1 min read)

### 1. Introduction

- This problem arises from various areas, for instance shape-optimization, fluid dy- namics, electrochemistry and electromagnetics.
- In particular, the free boundary (∂Ω \K) meets the fixed boundary (K) tangentially.
- Moreover, the solution has convex level sets.

### 2. Preliminaries

- The results in this subsection are more or less already known, but not proved in detail for this particular case.
- To finish up the preparations the authors show that the gradient is almost uniformly bounded on the boundary.
- Ω since the function used will be the same.

### 3. Beurling’s technique

- As mentioned earlier the authors revisit Beurling’s technique to prove their main result.
- The arguments that the authors will use are more or less the same as the proofs of Henrot and Shahgholian in [6] and [8].
- First, the authors give the steps of this technique and then, corresponding theorems.
- The class B is closed under intersection.
- Then there exists a solution of the free boundary problem (P) in a strong sense.

### 5. Generalizations and comments

- Let un denote the sequence of functions corresponding to the solution with the free boundary condition |∇un| = λn, and let Ωn be the corresponding domains.
- In the case of the arbritary p-Laplacian there are some small adjustments needed; the authors will have to replace the fact that |∇u|2 is subharmonic by [6, Lemma 2.1] and use that the therein defined operator Lu admits a maximum principle and that Lu|∇u|p ≥.
- Here the authors must use the general expression for the n-dimensional radial symmetric p-capacitary potential, i.e.
- The authors also remark that the C2+α-regularity will not be valid in the general case since the theorem used only applies to elliptic operators.
- As mentioned in the introduction, the result in this paper could probably be extended to results similar to those in [8] and [2] for their case.

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### Cites background from "A free boundary problem for the Lap..."

...About the literature on free boundary problem with a Bernoulli condition, we refer to [28, 29, 31, 32, 33, 35, 41, 42]....

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##### References

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### "A free boundary problem for the Lap..." refers background or methods or result in this paper

...The arguments that we will use are more or less the same as the proofs of Henrot and Shahgholian in [ 6 ] and [8]....

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...Proof. This is more or less the same arguments as in [ 6 ]....

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...limsup y!x |ru(y)| = 1. This result is already proved in [ 6 ]...

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...Proof. Even if this result is already proved in [ 6 ], we recall it in order to clarify the construction....

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...In order to prove that there is a minimal element in some sense (which we will soon see), we first need a theorem on sequences in B. The two following proofs can be found in [ 6 ]....

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