tailieunhanh - Báo cáo hóa học: " Research Article Interior Gradient Estimates for Nonuniformly Parabolic Equations II"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Interior Gradient Estimates for Nonuniformly Parabolic Equations II | Hindawi Publishing Corporation Boundary Value Problems Volume 2007 Article ID 35825 28 pages doi 2007 35825 Research Article Interior Gradient Estimates for Nonuniformly Parabolic Equations II Gary M. Lieberman Received 31 May 2006 Revised 6 November 2006 Accepted 9 November 2006 Recommended by Vincenzo Vespri We prove interior gradient estimates for a large class of parabolic equations in divergence form. Using some simple ideas we prove these estimates for several types of equations that are not amenable to previous methods. In particular we have no restrictions on the maximum eigenvalue of the coefficient matrix and we obtain interior gradient estimates for so-called false mean curvature equation. Copyright 2007 Gary M. Lieberman. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction A key step in the study of second-order quasilinear parabolic equations is establishing suitable a priori estimates for any solution of the equation. This fact is the theme of many books on the subject 1-5 and our focus here is on one particular such estimate a local pointwise gradient estimate for solutions of equations in divergence form ut divA X u Du B X u Du . The role of this divergence structure has been noted many times under varying hypotheses on the functions A and B see in particular 6 Sections and 3 Section 5 Section . Our current interest is deriving this estimate using a surprising variant detailed below of standard methods. Although this variant seems at first to be a purely technical modification we mention here two quite different types of estimates which follow from this variant and which appear to be new. First we derive a local gradient estimate for a class of equations which includes the parabolic false mean curvature 2 Boundary Value Problems equation that