Kaiser, Tobias Definability results for the Poisson equation. (English) Zbl 1137.35019 Adv. Geom. 6, No. 4, 627-644 (2006). 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 \(\mathbb{R}_{\text{an}}\). Reviewer: Aris Daniilidis (Bellaterra) Cited in 2 Documents 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 Keywords:Poisson equation; Laplace equation; subanalytic set; Pfaffian closure; o-minimal structure Citations:Zbl 0912.32007 PDFBibTeX XMLCite \textit{T. Kaiser}, Adv. Geom. 6, No. 4, 627--644 (2006; Zbl 1137.35019) Full Text: DOI References: [1] DOI: 10.1112/S0024609302001522 · Zbl 1036.31003 · doi:10.1112/S0024609302001522 [2] Comte J.-M., Illinois J. Math. 44 pp 884– (2000) [3] Denef L, Ann. of Math. 128 pp 79– (2) · Zbl 0693.14012 · doi:10.2307/1971463 [4] DOI: 10.2307/2000053 · Zbl 0662.03024 · doi:10.2307/2000053 [5] Lion J.-P, Ann. Inst. Fourier (Grenoble) 48 pp 755– (1998) [6] DOI: 10.1016/0168-0072(94)90048-5 · Zbl 0823.03018 · doi:10.1016/0168-0072(94)90048-5 [7] Remmert, Funktionentheorie. II. Springer pp 30002– (1991) [8] Speissegger, J. Reine Angew. Math. 508 pp 189– (1999) [9] van den Dries A., Ann. of Math. 140 (2) pp 183– (2015) [10] DOI: 10.1090/S0894-0347-96-00216-0 · Zbl 0892.03013 · doi:10.1090/S0894-0347-96-00216-0 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.