Advertisement

Analyticity of the semigroup in a degenerate case

Chapter
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Personalised recommendations