L p-theory for second-order elliptic operators with unbounded coefficients in an endpoint class

Abstract

The m-accretivity and m-sectoriality of the minimal and maximal realizations of second-order elliptic operators of the form \({Au=-{\rm div}(a \nabla u)+F\cdot \nabla u +Vu}\) in \({L^p(\mathbb{R}^N)}\) are shown, where the coefficients a, F and V are unbounded. The result may be regarded as an endpoint assertion of the previous result in Sobajima (J Evol Equ 12:957–971, 2012) and an improvement of that in Metafune et al. (Forum Math 22:583–601, 2010). Moreover, an L p-generalization of Kato’s self-adjoint problem in Kato (1981, Appendix 2) is discussed. The proof is based on Sobajima (J Evol Equ 12:957–971, 2012). As examples, the operators \({-\Delta \pm |x|^{\beta-1}x \cdot \nabla +c|x|^{\gamma}}\) are also dealt with, which are mentioned in Metafune et al. (Forum Math 22:583–601, 2010).

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Correspondence to Motohiro Sobajima.

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Sobajima, M. L p-theory for second-order elliptic operators with unbounded coefficients in an endpoint class. J. Evol. Equ. 14, 461–475 (2014). https://doi.org/10.1007/s00028-014-0223-9

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Mathematics Subject Classification (2000)

  • Primary 35J15
  • Secondary 47D06

Keywords

  • Second-order elliptic operators
  • unbounded coefficients
  • L p-generalization of Kato’s self-adjointness problem
  • m-Accretive operators in L p
  • m-Sectorial operators in L p