Does the Bi-Laplacian generate an analytic semigroup?

Question

It i well-known that the Laplace operator generate an analytic semigroup for example in $L^2(\Omega)$ on nice domains $\Omega \in \mathbb{R}^n$.

Now does also the Bi-Laplacian generate an analytic semigroup or at least a strongly continuous one? Can you provide references?

Answer 1

Yes, the bi-Laplacian with Dirichlet boundary conditions generates an analytic semigroup on $L^p(\Omega), \; p\in (1,\infty)$. For $p=2$, you can show that the operator is self-adjoint. A general result was proved in Theorem 5.6, pp. 189 in

Hiroki Tanabe, Functional Analytic Methods for Partial Differential Equations, vol 204 of Monographs and Textbooks in Pure and Applied Mathematics, New York, 1997.