On the behavior of the free boundary for a one-phase Bernoulli problem with mixed boundary conditions

Giovanni Gravina; Giovanni Leoni

doi:10.3934/cpaa.2020215

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2020, Volume 19, Issue 10: 4853-4878. Doi: 10.3934/cpaa.2020215

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On the behavior of the free boundary for a one-phase Bernoulli problem with mixed boundary conditions

Giovanni Gravina^1, and
Giovanni Leoni^2, ,

1.
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
2.
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA

^* Corresponding author
^* Corresponding author

Received: November 2019

Revised: June 2020

Published: July 2020

The authors acknowledge the Center for Nonlinear Analysis (NSF PIRE Grant No. OISE- 0967140) where part of this work was carried out. The research of G. Gravina and G. Leoni was partially funded by the National Science Foundation under Grants No. DMS-1412095 and DMS- 1714098. G. Gravina also acknowledges the support of the research support programs of Charles University: PRIMUS/19/SCI/01 and UNCE/SCI/023

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Abstract

This paper is concerned with the study of the behavior of the free boundary for a class of solutions to a two-dimensional one-phase Bernoulli free boundary problem with mixed periodic-Dirichlet boundary conditions. It is shown that if the free boundary of a symmetric local minimizer approaches the point where the two different conditions meet, then it must do so at an angle of $ \pi/2 $.

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Mathematics Subject Classification: 35R35.

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Figure 1. The free boundary meets the Dirichlet fixed boundary tangentially

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Figure 2. The free boundary hits the point $ (- \lambda/2, \gamma) $ at an angle of $ \pi/2 $

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Figure 3. Qualitative behavior of the free boundary near the fixed boundary

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