BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
نویسنده
چکیده
Maximum principles are among the most powerful and widely used analytic tools in the study of second-order linear and nonlinear elliptic and parabolic equations. They enable us to obtain valuable information about (real valued) solutions of differential equations and inequalities (such as a priori pointwise estimates, and uniqueness and stability results) without the need to know in advance the solutions explicitly, or without even knowing a priori the existence of such solutions. As a matter of fact, in many cases, the maximum principle (or MP for brevity) is an essential ingredient in proving also existence theorems. Moreover, MPs are closely related to some well known important qualitative properties of solutions of such equations, e.g., Harnack inequalities, comparison principles and tangency theorems, Phragmèn-Lindelöf principles, removability of isolated singularities, and Liouville theorems. As an illustration of the MP, we mention the everyday fact that a body with a prescribed (time-independent) boundary temperature attains the highest temperature of a steady-state temperature on its boundary. The present article briefly surveys the exciting field of maximum principles with emphasis on the content and features of the book under review (which will be denoted by [PS]).
منابع مشابه
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
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تاریخ انتشار 2009