Analyticity of semigroups generated by higher order elliptic operators in spaces of bounded functions on \(C^{1}\) domains. (English) Zbl 1380.35079

Suzuki, Takuya

Adv. Differ. Equ. 22, No. 9-10, 593-620 (2017).

Under consideration is the problem \[ \lambda u-\Delta^{2}u=f, \quad u|_{\Gamma}=\partial_{\nu}u|_{\Gamma}=0,\quad x\in G\subset {\mathbb R}^{n}, \] where \(G\) is a \(C^{1}\)-domain (possibly unbounded) and \(\nu\) is the unit normal to \(\Gamma=\partial G\). The following resolvent estimate is proven for solutions \(u\in W_{p,\mathrm{loc}}^{2}(G)\cap W_{\infty}^{1}(G)\) \((p>n)\): \[ |\lambda|\|u\|_{L_{\infty}(G)}+|\lambda|^{3/4}\|\nabla u\|_{L_{\infty}(G)}\leq c\|f\|_{L_{\infty}(G)},\; \] where the constant \(c\) is independent of \(\lambda\in S=\{\lambda: |\mathrm{arg} \lambda|\leq \pi-\varepsilon\}\), \(0<\varepsilon<\pi\).

Reviewer: S. G. Pyatkov (Khanty-Mansiysk)

MSC:

35J40	Boundary value problems for higher-order elliptic equations
35J30	Higher-order elliptic equations
47A10	Spectrum, resolvent
31B30	Biharmonic and polyharmonic equations and functions in higher dimensions

Keywords:

biharmonic operator; elliptic problem; resolvent estimate