Mohammad Almahameed
In this paper we consider several types of differential equations and discuss the maximum principle for them. In general, the maximum principle tells us that the maximum value of the function, which is a solution of a differential equation, is attained at the boundary of the region. In this paper, we deal with elliptic equations. The most important and easy equation is the Laplace equation. The homogeneous version of Laplace’s equation is ?u = 0. It is often written with minus sign since the (delta-operator) with this sign becomes strict monotone operator in the operator theory, which means that it has a unique solution. The non-homogeneous version of Laplace’s equation ?u = f is called Poisson’s equation. It is convenient to include a minus sign here because ? is a negative definite operator. The Laplace and Poisson equations, and their generalizations, arise in many different contexts. 2010 Math. Subject Classification: 35R45, 35R50
Differential Equations, Maximum principles