Estimates of the derivatives for parabolic operators with unbounded coefficients

By M. Bertoldi, Luca Lorenzi. 2005

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We consider a class of second-order uniformly elliptic operators A with unbounded coefficients in R N . Using a Bernstein approach we provide several uniform estimates for the semigroup T(t) generated by the realization of the operator A in the space of all bounded and continuous or Holder continuous functions in R N . As a consequence, we obtain optimal Schauder estimates for the solution to both the elliptic equation λu - Au = f (λ > 0) and the nonhomogeneous Dirichlet Cauchy problem D t u = Au + g. Then, we prove two different kinds of pointwise estimates of T(t) that can be used to prove a. Liouville-type theorem. Finally, we provide sharp estimates of the semigroup T(t) in weighted L p -spaces related to the invariant measure associated with the semigroup.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bc52f968d4cfc365dc806c3e31be81ab53b5a1c5