We prove a maximal regularity result for operators corresponding to rotation invariant symbols (in space) which are inhomogeneous in space and time. Symbols of this type frequently arise in the treatment of half-space models for (free) boundary-value problems. The result is obtained by extending the Newton polygon approach to variables living in complex sectors and combining it with abstract results on the $ \mathcal{H}^\infty $ -calculus and $ \mathcal{R} $ -bounded operator families. As an application, we derive maximal regularity for the linearized Stefan problem with Gibbs-Thomson correction.

We investigate the Bi-Laplacian with Wentzell boundary conditions in a bounded domain $$\Omega \subseteq \mathbb {R}^d$$Ω⊆Rd with Lipschitz boundary $$\Gamma $$Γ . More precisely, using form methods, we show that the associated operator on the ground space $$L^2(\Omega )\times L^2(\Gamma )$$L2(Ω)×L2(Γ) 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.