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Existence of quadrature surfaces for uniform density supported by a segment. (English) Zbl 1160.35380

Given are two strictly positive constants \(a\) and \(k.\) It is shown that if \(a \geq 3.92k\) then there exists an open and bounded set \(\Omega\) in \(\mathbb{R}^2\) which contains strictly the line segment \(C\) \((C = [-1,1] \times \{0\})\) such that the following overdetermined problem has a solution \(-\Delta u = a\delta_C\) in \(\Omega,\) \(u = 0\) and \(-\frac{\partial u}{\partial \nu} = k\) on \(\partial \Omega.\) Here \(\nu\) is the outward normal vector to \(\partial \Omega\) and \(\delta_C\) is the uniform density supported by \(C.\)

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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