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

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 $$\lambda\in S=\{\lambda: |\mathrm{arg} \lambda|\leq \pi-\varepsilon\}$$, $$0<\varepsilon<\pi$$.

### 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
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