Elliptic fourth-order operators with Wentzell boundary conditions on Lipschitz domains

For bounded domains \(\Omega \) $\mathrm{\Omega }$ with Lipschitz boundary \(\Gamma \) $\mathrm{\Gamma }$ , we investigate boundary value problems for elliptic operators with variable coefficients of fourth-order subject to Wentzell (or dynamic) boundary conditions. Using form methods, we begin by showing general results for an even wider class of operators of type \(\begin{aligned} {\mathcal {A}}=\begin{pmatrix} B^*B & 0 \\ -{\mathscr {N}}^{\mathfrak {b}}B & \gamma \end{pmatrix}, \end{aligned}\) $$\begin{array}{c}\hfill \mathcal{A}=\left(\begin{array}{cc}{B}^{\ast }B& 0\\ -{\mathcal{N}}^{\mathfrak{b}}B& \gamma \end{array}\right),\end{array}$$ where B is associated to a quadratic form \({\mathfrak {b}}\) $\mathfrak{b}$ and \({\mathscr {N}}^{{\mathfrak {b}}}\) ${\mathcal{N}}^{\mathfrak{b}}$ an abstractly defined co-normal Neumann trace. Even in this general setting, we prove generation of an analytic semigroup on the product space \({\mathcal {H}}:=L^2(\Omega ) \times L^2(\Gamma )\) $\mathcal{H}:=L^2(\mathrm{\Omega })\times L^2(\mathrm{\Gamma })$ . Using recent results concerning weak co-normal traces, we apply our abstract theory to the elliptic fourth-order case and are able to fully characterize the domain in terms of Sobolev regularity for operators in divergence form \(B=-\mathop {{div} }Q \nabla \) $B=-\mathit{div}Q\mathrm{\nabla }$ with \(Q \in C^{1,1}({\overline{\Omega }},{\mathbb {R}}^{d\times d}),\) $Q\in C^{1,1}(\overline{\mathrm{\Omega }},{\mathbb{R}}^{d\times d}),$ also obtaining Hölder-regularity of solutions. Finally, we also discuss asymptotic behavior and (eventual) positivity.