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Sectoriality of degenerate elliptic operators via \(p\)-ellipticity. (English) Zbl 1506.35092

Summary: Let \(\Omega\subset\mathbb{R}^d\) be open and \(c_{kl}\in L_{\infty}(\Omega,\mathbb{C})\) with \(\mathrm{Im}c_{kl}=\mathrm{Im} c_{lk}\) for all \(k,l\in\{1,\ldots,d\}\). Assume that \(C=(c_{kl})_{1\leq k,l\leq d}\) satisfies \((C(x)\xi,\xi)\in\Sigma_{\theta}\) for all \(x\in\Omega\) and \(\xi\in\mathbb{C}^d\), where \(\Sigma_{\theta}\) is the closed sector with vertex 0 and semi-angle \(\theta\) in the complex plane. We emphasize that \(\Omega\) is an arbitrary domain and \(C\) need not be symmetric. We show that \(C\) is (degenerate) \(p\)-elliptic for all \(p\in (1,\infty)\) with \(|1-\frac{2}{p}|<\cos\theta\) in the sense of Carbonaro and Dragičević. As a consequence, we obtain the consistent holomorphic extension for the \(C_0\)-semigroup generated by the second-order differential operator in divergence form associated with \(C\). The core property for this operator is also investigated.

MSC:

35J70 Degenerate elliptic equations
35J15 Second-order elliptic equations
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