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One-dimensional degenerate elliptic operators on \(L_{\!p}\)-spaces with complex coefficients

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Abstract

Let \(c :\mathbb {R}\rightarrow \mathbb {C}\) be a bounded Lipschitz continuous function which takes values in a sector. We consider the divergence form operator \(A = - \frac{d}{dx} \, c \, \frac{d}{dx}\) in \(L_2(\mathbb {R})\). We characterize for which \(p \in [1,\infty )\) the semigroup generated by \(-A\) extends consistently to a contraction \(C_0\)-semigroup on \(L_p(\mathbb {R})\) and for those \(p\) we characterize when \(C_c^\infty (\mathbb {R})\) is a core for the generator in \(L_p(\mathbb {R})\).

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Acknowledgments

We wish to thank the referee for his comments and for suggesting that the ‘if’-part in Proposition 1.4 is valid for merely an \(L_\infty \)-coefficient. This is now in Proposition 1.2. Part of this work is supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand.

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  1. Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand

    Tan Duc Do & A. F. M. ter Elst

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  1. Tan Duc Do
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  2. A. F. M. ter Elst
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Correspondence to Tan Duc Do.

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Communicated by Markus Haase.

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Do, T.D., ter Elst, A.F.M. One-dimensional degenerate elliptic operators on \(L_{\!p}\)-spaces with complex coefficients. Semigroup Forum 92, 559–586 (2016). https://doi.org/10.1007/s00233-015-9721-5

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