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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References
Agmon, S.: Lectures on Elliptic Boundary Value Problems. AMS Chelsea Publishing, Providence (2010)
Arendt, W., ter Elst, A.F.M.: Sectorial forms and degenerate differential operators. J. Oper. Theory 67, 33–72 (2012)
Auscher, P., Barthélemy, L., Bénilan, P., Ouhabaz, E.M.: Absence de la \(L^\infty \)-contractivité pour les semi-groupes associés aux opérateurs elliptiques complexes sous forme divergence. Potential Anal. 12, 160–189 (2000)
Auscher, P., McIntosh, A., Tchamitchian, P.: Heat kernels of second order complex elliptic operators and their applications. J. Funct. Anal. 152, 22–73 (1998)
Campiti, M., Metafune, G., Pallara, D.: Degenerate self-adjoint evolution equations on the unit interval. Semigroup Forum 57, 1–36 (1998)
Cialdea, A., Maz’ya, V.: Criterion for the \(L^p\)-dissipativity of second order differential operators with complex coefficients. J. Math. Pures Appl. 84(9), 1067–1100 (2005)
Davies, E.B.: \(L^1\) properties of second order elliptic operators. Bull. Lond. Math. Soc. 17, 417–436 (1985)
ter Elst, A.F.M., Robinson, D.W., Sikora, A.: Flows and invariance for elliptic operators. J. Aust. Math. Soc. 90, 317–339 (2011)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften, vol. 224, 2nd edn. Springer, Berlin (1983)
Hamza, M. M.: Détermination des formes de Dirichlet sur \(\mathbb{R}^n\). Thèse 3e cycle, Orsay (1975)
Kato, T.: Perturbation theory for linear operators. Grundlehren der Mathematischen Wissenschaften, vol. 132, 2nd edn. Springer, Berlin (1980)
Kato, T.: Remarks on the selfadjointness and related problems for differential operators. Spectral Theory of Differential Operators. North-Holland Mathematics Studies, vol. 55, pp. 253–266. North-Holland, Amsterdam (1981)
Liskevich, V.: Essential self-adjointness of semibounded elliptic operators of second order. Ukrain. Mat. Zh. 41 (1989), 710–716. Translation in Ukrainian Math. J. 41(5), 615–619 (1989)
Lumer, G., Phillips, R.S.: Dissipative operators in a Banach space. Pac. J. Math. 11, 679–698 (1961)
Metafune, G., Pallara, D., Rabier, P.J., Schnaubelt, R.: Uniqueness for elliptic operators on \(L^p(\mathbb{R}^N)\) with unbounded coefficients. Forum Math. 22, 583–601 (2010)
Ouhabaz, E.-M.: Analysis of Heat Equations on Domains. London Mathematical Society Monographs Series, vol. 31. Princeton University Press, Princeton (2005)
Röckner, M., Wielens, N.: Dirichlet forms - closability and change of speed measure. Infinite-Dimensional Analysis and Stochastic Processes (Bielefeld, 1983). Research Notes in Mathematics, vol. 124, pp. 119–144. Pitman, Boston (1985)
Voigt, J.: One-parameter semigroups acting simultaneously on different \(L_p\)-spaces. Bull. Soc. R. Sci. Liège 61, 465–470 (1992)
Wong-Dzung, B.: \(L^p\)-Theory of degenerate-elliptic and parabolic operators of second order. Proc. R. Soc. Edinburgh Sect. A 95, 95–113 (1983)
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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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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DOI: https://doi.org/10.1007/s00233-015-9721-5