We investigate the Bi-Laplacian with Wentzell boundary conditions in a bounded domain \(\Omega \subseteq \mathbb {R}^d\) with Lipschitz boundary \(\Gamma \). More precisely, using form methods, we show that the associated operator on the ground space \(L^2(\Omega )\times L^2(\Gamma )\) has compact resolvent and generates a holomorphic and strongly continuous real semigroup of self-adjoint operators. Furthermore, we give a full characterization of the domain in terms of Sobolev spaces, also proving Hölder regularity of solutions, allowing classical interpretation of the boundary condition. Finally, we investigate spectrum and asymptotic behavior of the semigroup, as well as eventual positivity.

我们研究具有Lipschitz边界\（\ Gamma \）的有界域\（\ Omega \ subseteq \ mathbb {R} ^ d \）中具有Wentzell边界条件的Bi-Laplacian 。更准确地说，使用形式方法，我们证明了地面空间\（L ^ 2（\ Omega）\ times L ^ 2（\ Gamma）\）上的关联算符具有紧凑的分解子，并生成一个全纯且强连续的实半群自伴运算符。此外，我们根据Sobolev空间对域进行了全面表征，还证明了解的Hölder正则性，从而可以对边界条件进行经典解释。最后，我们研究了半群的频谱和渐近行为，以及最终的积极性。