A minimization problem related to the principal frequency of the $p$-Bilaplacian with coupled Dirichlet–Neumann boundary conditions

Abstract

For each fixed integer $N\geq 2$ let $\Omega\subset\mathbb{R}^{N}$ be an open, bounded and convex set with smooth boundary. For each real number $p\in(1,\infty)$ define $$M(p;\Omega)=\inf_{u\in {\mathcal{W}}_C^{2,\infty}(\Omega)\setminus\{0\}}\dfrac{\displaystyle\int_\Omega (\exp(|\Delta u|^p)-1)\;dx}{\displaystyle \int_\Omega(\exp(|u|^p)-1)\;dx}\,,$$ where ${\mathcal{W}}_C^{2,\infty}(\Omega):=\cap_{1<p<\infty}\{u\in W_0^{2,p}(\Omega):\;\Delta u\in L^\infty(\Omega)\}$. We show that if the radius of the largest ball which can be inscribed in $\Omega$ is strictly larger than a constant which depends on $N$ then $M(p;\Omega)$ vanishes while if the radius of the largest ball which can be inscribed in $\Omega$ is strictly less than $1$ then $M(p;\Omega)$ is a positive real number. Moreover, in the latter case when $p$ is large enough we can identify the value of $M(p;\Omega)$ as being the principal frequency of the $p$-Bilaplacian on $\Omega$ with coupled Dirichlet–Neumann boundary conditions.

Keywords

• $p$-bilaplacian
• principal frequency
• dirichlet–neumann boundary conditions