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Kolmogorov equations in Rd with unbounded coefficients

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

Abstract

In this chapter we are concerned with the following class of second order elliptic operators
$$ \mathcal{L}_0 (x,D) = \sum\limits_{i,j = 1}^d {a_{ij} (x)D_{ij} + \sum\limits_{i = 1}^d {b_i (x)D_{i, } x\varepsilon \mathbb{R}^d } } $$
The vector field b = (b,...,b d) : ℝd → ℝd is of class C 3 and the matrix a(x) = [a ij(x)] is symmetric, strictly positive and of class C 3, so that it can be written as a(x) = ½σxσ* x, x∈ℝd, for some function σ : ℝd → ℒ(ℝd) of class C 2 (in fact, we can take σ = √a). Both b and a are assumed to have polynomial growth and 6 enjoys some dissipativity conditions which will be described in more details later on. Our aim is to prove the existence and the uniqueness of solutions for the elliptic and the parabolic problems associated with the operator ℒ0. Moreover, we want to give optimal regularity results in the space of bounded Holder continuous functions (Schauder type estimates).

Keywords

Variation Equation Parabolic Problem Previous Proposition KOLMOGOROV Equation Gronwall Lemma 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

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