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 Harnack Inequalities: An Introduction Moritz Kassmann | Hindawi Publishing Corporation Boundary Value Problems Volume 2007 Article ID 81415 21 pages doi 2007 81415 Research Article Harnack Inequalities An Introduction Moritz Kassmann Received 12 October 2006 Accepted 12 October 2006 Recommended by Ugo Pietro Gianazza The aim of this article is to give an introduction to certain inequalities named after Carl Gustav Axel von Harnack. These inequalities were originally defined for harmonic functions in the plane and much later became an important tool in the general theory of harmonic functions and partial differential equations. We restrict ourselves mainly to the analytic perspective but comment on the geometric and probabilistic significance of Har-nack inequalities. Our focus is on classical results rather than latest developments. We give many references to this topic but emphasize that neither the mathematical story of Harnack inequalities nor the list of references given here is complete. Copyright 2007 Moritz Kassmann. 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. Carl Gustav Axel von Harnack On May 7 1851 the twins Carl Gustav Adolf von Harnack and Carl Gustav Axel von Harnack are born in Dorpat which at that time is under German influence and is now known as the Estonian university city Tartu. Their father Theodosius von Harnack 1817-1889 works as a theologian at the university. The present article is concerned with certain inequalities derived by the mathematician Carl Gustav Axel von Harnack who died on April 3 1888 as a Professor of mathematics at the Polytechnikum in Dresden. His short life is devoted to science in general mathematics and teaching in particular. For a mathematical obituary including a complete list of Harnack s C. G. Axel von Harnack 1851-1888 publications we refer the reader to 1 photograph courtesy of Professor em. .
