The torsion problem of the p-Bilaplacian

Abstract

For each bounded and open set $\Omega \subset R^N$ ($N\ge 2$) with smooth boundary denoted by $\partial \Omega$ and each real number $p\in (1,\infty )$ we analyze the torsion problem of the $p$-Bilaplacian, namely $\Delta \left({\left|\Delta u\right|}^{p-2}\Delta u\right)=1$ in $\Omega$ with $u=\Delta u=0$ on $\partial \Omega$. Firstly, we show that for each $p\in (1,\infty )$ the problem has a unique weak solution $u_p$. Secondly, we prove that $u_p$ converges uniformly, as $p\to \infty$, in $C^1\left(\overline{\Omega }\right)$ to a certain function, say $v_2$, which is exactly the unique solution of the problem $-\Delta u=1$ in $\Omega$ with $u=0$ on $\partial \Omega$. Moreover, for each real number $q\in [1,\infty )$, $\Delta u_p$ converges strongly to $\Delta v_2$ in $L^q\left(\Omega \right)$, as $p\to \infty$. Next, we show that each solution $u_p$ is also a solution for the minimization problem $T(p;\Omega )≔\underset{u\in X_p\left(\Omega \right)\setminus \left\{0\right\}}{inf}\frac{\frac{1}{\left|\Omega \right|}{\int }_{\Omega }{\left|\Delta u\right|}^{p}dx}{{\left(\frac{1}{\left|\Omega \right|}{\int }_{\Omega }udx\right)}^p}$ where $X_p\left(\Omega \right)≔\{u\in W^{2,p}\left(\Omega \right)\cap W_0^{1,p}\left(\Omega \right):u\left(x\right)\ge 0,a.e.x\in \Omega \}$. Further, we show that the function $(1,\infty )\ni p\mapsto T(p;\Omega )$ is strictly increasing provided that $\Omega$ is a convex and bounded open set for which ${\left|\Omega \right|}^{-1}{\int }_{\Omega }{v}_{2}dx$ is small. Finally, using this monotonicity result, we give an alternative variational characterization of the constant $T(p;\Omega )$ when ${\left|\Omega \right|}^{-1}{\int }_{\Omega }{v}_{2}dx$ is small. That last variational characterization fails to hold true when ${\left|\Omega \right|}^{-1}{\int }_{\Omega }{v}_{2}dx>1$.