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Definability results for the Poisson equation. (English) Zbl 1137.35019

Given a simply connected bounded semianalytic subset \(\Omega \) of \(\mathbb{R}^{2}\) with an analytical smooth boundary \(\partial \Omega \) and two bounded continuous subanalytic functions \(f:\Omega \rightarrow \mathbb{R}\) and \(g:\partial \Omega \rightarrow \mathbb{R}\) the author shows that the (unique) solution \(u:\overline{\Omega}\rightarrow \mathbb{R} \) to the Poisson equation \(\triangle u=f\) in \(\Omega \) subject to the boundary condition \(u=g\) on \(\partial \Omega \) is definable in the o-minimal structure \(\mathbb{R}_{\text{an},\exp }\) (that is, the o-minimal structure generated by the globally subanalytic sets and the exponential mapping). Existence of solutions is based on the fact that the function \(f\) belonging to the polynomially bounded structure \(\mathbb{R}_{\text{an}}\) of globally subanalytic sets is Hölder continuous, while uniqueness is a consequence of the maximum principle.) Using a result of J.-M. Lion and J.-P. Rolin [Ann. Inst. Fourier 48, 755–767 (1998; Zbl 0912.32007)], the author establishes at first the result in the particular case when \(\Omega \) is the unit ball \(B(0,1)\), and concludes by the definability (in \(\mathbb{R}_{\text{an}}\)) of the Riemann mapping that maps \(\overline{\Omega}\) to \(\overline{B}(0,1).\) In particular case \(f=0\) (Laplace equation) it is known that the solution \(u\) is analytic in \(\Omega \), but eventually without analytic extension at the boundary \(\partial \Omega \). Under necessary assumptions on the boundary function \(g\) the author gives a dichotomy result stating that either \(u\) is analytic at \(x\in\partial \Omega \) (thus \(u\) is definable in \(\mathbb{R}_{\text{an}}\)) or \(u\) is definable in \(\mathbb{R}_{\text{an},\exp}\) but not in \(\mathbb{R}_{\text{an}}\).

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
14P15 Real-analytic and semi-analytic sets
32B20 Semi-analytic sets, subanalytic sets, and generalizations

Citations:

Zbl 0912.32007
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References:

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