Analyticity of the semigroup in a degenerate case

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


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.


Degenerate Case Analytic Semigroup Stochastic Partial Differential Equation Quadratic Growth Order Differential Operator 
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© Springer-Verlag Berlin Heidelberg 2001

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