Non-uniqueness in a free boundary problem

Bennewitz, Björn

doi:10.4171/rmi/547

Cited by 3 publications

(2 citation statements)

References 12 publications

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“…In this subsection we give a brief introduction to a technique used by Beurling in [11] and in [25] as well. To this end, recall that K is a convex domain and let…”

Section: A Technique Of Beurlingmentioning

confidence: 99%

On a Bernoulli-type overdetermined free boundary problem

2021

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In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in [25] to A-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modelled on the p-Laplace equation for a fixed 1 < p < ∞. In particular, we show that if K is a bounded convex set satisfying the interior ball condition and c > 0 is a given constant, then there exists a unique convex domain Ω with K ⊂ Ω and a function u which is A-harmonic in Ω \ K, has continuous boundary values 1 on ∂K and 0 on ∂Ω, such that |∇u| = c on ∂Ω. Moreover, ∂Ω is C 1,γ for some γ > 0, and it is smooth provided A is smooth in R n \ {0}. We also show that the super level sets {u > t} are convex for t ∈ (0, 1).

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“…In this subsection we give a brief introduction to a technique used by Beurling in [11] and in [25] as well. To this end, recall that K is a convex domain and let…”

Section: A Technique Of Beurlingmentioning

confidence: 99%

On a Bernoulli-type overdetermined free boundary problem

2021

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show abstract

“…Finally, using Lemmas 2.1 and 2.2 we conclude that Lemma 3.1 holds for u as well. Beurling. In this subsection we give a brief introduction to a technique used by Beurling in [11] and in [26] as well. To this end, recall that K is a convex domain and let…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

On a Bernoulli-type overdetermined free boundary problem

Akman¹,

Banerjee²,

Garcia³

2019

Preprint

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In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in [26] to A-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modelled on the p-Laplace equation for a fixed 1 < p < ∞. In particular, we show that if K is a bounded convex set satisfying the interior ball condition and c > 0 is a given constant, then there exists a unique convex domain Ω with K ⊂ Ω and a function u which is A-harmonic in Ω \ K, has continuous boundary values 1 on ∂K and 0 on ∂Ω, such that |∇u| = c on ∂Ω. Moreover, ∂Ω is C 1,γ for some γ > 0, and it is smooth provided A is smooth in R n \ {0}. We also show that the super level sets {u > t} are convex for t ∈ (0, 1).

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Applications of Boundary Harnack Inequalities for p Harmonic Functions and Related Topics

Lewis

2012

Lecture Notes in Mathematics

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Non-uniqueness in a free boundary problem

Abstract: We show that a result of Lewis and Vogel on uniqueness in a free boundary problem for the p-Laplace operator is sharp in two dimensions.

Cited by 3 publications

References 12 publications

On a Bernoulli-type overdetermined free boundary problem

On a Bernoulli-type overdetermined free boundary problem

On a Bernoulli-type overdetermined free boundary problem

Applications of Boundary Harnack Inequalities for p Harmonic Functions and Related Topics

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