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A free boundary problem for the Laplacian with a constant Bernoulli-type boundary condition. (English) Zbl 1123.35092

Summary: 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 super-solutions and a Perron argument. We try to generalize here a previous work of A. Henrot and H. Shahgholian [Trans. Am. Math. Soc. 354, No. 6, 2399–2416 (2002; Zbl 0988.35174) and Interfaces Free Bound. 6, No. 1, 81–103 (2004; Zbl 1050.35148)]. We extend these results in different directions.

MSC:

35R35 Free boundary problems for PDEs
35J25 Boundary value problems for second-order elliptic equations
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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