一类非线性散度型椭圆方程的最大值原理

Maximum Principles for a Class of Nonlinear Elliptic Equations in Divergence Form

本文研究了一类非线性散度型椭圆方程解的函数的最大值原理.最大值原理在偏微分方程中对于解的存在性、唯一性和先验界的估计等问题的研究具有非常重要的作用.本文构造了带有梯度项的P-函数,然后利用Hopf最大值原理和所构造的P-函数,获得了该方程在Dirichlet边值条件和Robin边值条件下的最大值原理.最后,通过一个实例验证了文中所获得的最大值原理的有效性.

In this paper, we discuss the maximum principles for solutions of a class of nonlinear elliptic equations in divergence form. The maximum principle plays a very important role in the problems such as the existence, uniqueness and priori estimates of the solutions of partial differential equations. The main idea of this paper is to construct P-function with gradient term. By employing Hopf＇s maximum principle and P-function, we obtain some maximum principles for such equations subject to Dirichlet or Robin boundary conditions. Finally, we verify the effectiveness of the maximum principles by an numerical example.