In this paper we study the global boundedness for solutions to a class of possibly degenerate parabolic equations by De-Giorgi's iteration. As applications, show existence weak stochastic differential with singular diffusion and drift coefficients. Moreover, Markov selection theorem Krylov [8], also establish associated strong family.

Mohammed Kbiri Alaoui Department of Mathematics, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia Correspondence should be addressed to Mohammed Kbiri Alaoui, mka [email protected] Received 31 March 2011; Accepted 28 July 2011 Academic Editor: Mihai Putinar Copyright q 2011 Mohammed Kbiri Alaoui. This is an open access article distributed under the Creative Commons Attribution License, which...

These notes contain a short exposition of selected results about parabolic equations: Schauder estimates for linear parabolic equations with Hölder coefficients, some existence, uniqueness and regularity results for viscosity solutions of fully nonlinear parabolic equations (including degenerate ones), the Harnack inequality for fully nonlinear uniformly parabolic equations. MSC. 35K55, 35D40, ...

We derive and analyze monotone difference-quadrature schemes for Bellman equations of controlled Lévy (jump-diffusion) processes. These equations are fully non-linear, degenerate parabolic integro-PDEs interpreted in the sense of viscosity solutions. We propose new “direct” discretizations of the non-local part of the equation that give rise to monotone schemes capable of handling singular Lévy...

We derive and analyze monotone difference-quadrature schemes for Bellman equations of controlled Lévy (jump-diffusion) processes. These equations are fully non-linear, degenerate parabolic integro-PDEs interpreted in the sense of viscosity solutions. We propose new “direct” discretizations of the non-local part of the equation that give rise to monotone schemes capable of handling singular Lévy...

We present nonnegativity-preserving finite element schemes for a general class of thin film equations in multiple space dimensions. The equations are fourth order degenerate parabolic, and may contain singular terms of second order which are to model van der Waals interactions. A subtle discretization of the arising nonlinearities allows us to prove discrete counterparts of the essential estima...