On an Isoperimetric Inequality for the First Eigenvalue of a Boundary Value Problem
Miriam Bareket · SIAM Journal on Mathematical Analysis · 1977
Let D be a two-dimensional simply connected bounded domain whose boundary $\partial D$ consists of a finite number of regular arcs. This paper suggests that for all such domains D of the same area A, the circle yields the maximum value for the first eigenvalue $\lambda _1 $ of the problem: \[ \begin{gathered} \Delta u + \lambda u = 0 \quad {\text{in }}D, \hfill \\ \frac{{\partial u}}{{\partial n}} = Zu \qquad {\text{on }}\partial D. \hfill \\ \end{gathered} \] Here ${\partial / {\partial n}}$ denotes differentiation with respect to the exterior normal to D and Z is a positive constant. This isoperimetric property of $\lambda _1 $ is proved for any $Z > 0$ under certain assumptions on the circumference, and the local extremism property is shown for certain values of the parameter.