## 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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