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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
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Full Text: Euclid