We prove some results about the first Steklov eigenvalue $d_1$ of the biharmonic operator in bounded domains. Firstly, we show that Fichera's principle of duality \cite{fichera} may be extended to a wide class of nonsmooth domains. Next, we study the optimization of $d_1$ for varying domains: we disprove a long-standing conjecture, we show some new and unexpected features and we suggest some challenging problems. Finally, we prove several properties of the ball.
Bucur, D., Ferrero, A., Gazzola, F. (2009). On the first eigenvalue of a fourth order Steklov problem. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 35(1), 103-131 [10.1007/s00526-008-0199-9].
On the first eigenvalue of a fourth order Steklov problem
FERRERO, ALBERTO;
2009
Abstract
We prove some results about the first Steklov eigenvalue $d_1$ of the biharmonic operator in bounded domains. Firstly, we show that Fichera's principle of duality \cite{fichera} may be extended to a wide class of nonsmooth domains. Next, we study the optimization of $d_1$ for varying domains: we disprove a long-standing conjecture, we show some new and unexpected features and we suggest some challenging problems. Finally, we prove several properties of the ball.
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