Poisson Stochastic Process and Basic Schauder and Sobolev Estimates in the Theory of Parabolic Equations (Short Version)
We show among other things how knowing Schauder or Sobolev-space estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs for equations with coefficients depending only on time variable with the same constants as in the case of the one-dimensional heat equation. The method is quite general and is based on using the Poisson stochastic process. It also applies to equations involving non-local operators. It looks like no other method is available at this time and it is a very challenging problem to find a purely analytic approach to proving such results. We only give examples of applications of our results. Their proofs will appear elsewhere.
KeywordsSchauder estimates Sobolev-space estimates Multidimensional parabolic equations Poisson process
2010 Mathematics Subject Classification35K10 35K15
The article is based on the talk given by the first author at the international conference “Stochastic Partial Differential Equations and Related Topics” October 10–14, 2016, Bielefeld University. The opportunity to give this talk is greatly appreciated.
- 5.Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and quasi-linear parabolic equations, Nauka, Moscow (1967), in Russian; English translation: American Mathematical Society, Providence, RI (1968)Google Scholar