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Analyticity of the semigroup in a degenerate case

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1762)

Abstract

We consider the following class of second order differential operators d d

$$ \mathcal{L}_O (x,D) = \sum\limits_{i,j = 1}^d {a_{ij} (x)D_{ij + } \sum\limits_{i = 1}^d {b_i (x)D_i , x \in \mathbb{R}^d ,} } $$

where a(x) is a positive semi-definite symmetric matrix which has quadratic growth and b(x) is a vector field of class C 2 which has linear growth. We assume that the mapping a : ℝd → ℒ(ℝd) is also of class C 2 with bounded second derivatives, so that it can be written as a(x) = ½σxσ* x, x∈ℝd, for some matrix valued function σ : ℝd → ℒ(ℝd) which is Lipschitz-continuous (for a proof of this fact see [66] and also [114]. Here we assume further regularity for o; namely we assume that a can be factorized by some a which is twice differentiable with bounded derivatives.

Keywords

  • Degenerate Case
  • Analytic Semigroup
  • Stochastic Partial Differential Equation
  • Quadratic Growth
  • Order Differential Operator

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 2001 Springer-Verlag Berlin Heidelberg

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(2001). Analyticity of the semigroup in a degenerate case. In: Cerrai, S. (eds) Second Order PDE’s in Finite and Infinite Dimension. Lecture Notes in Mathematics, vol 1762. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45147-1_4

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  • DOI: https://doi.org/10.1007/3-540-45147-1_4

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42136-8

  • Online ISBN: 978-3-540-45147-1

  • eBook Packages: Springer Book Archive