1.1.The Maximum Principle 9 Theorem 1.1.2.Let be a bounded domain in R"and L be given by (1.1,for somei(()satisfving i and (1.1.2).Suppose that C()()satisfies Lu0 in Then,u attains onon its nonnegative marimum in S;i.e., mgxw≤xu+, Proof.We first consider the special case Lu >0 in If u has a local nonnegative maximum at a point o in then u(o)0,Vu(o)=0, and the Hessian matrix ((is negative semi-definite.By (1.1.),the matrix (a(xo))is positive definite.Then, Lu(xo）=(aaju+b:8,u+cu)(xo)≤0. This leads to a contradiction.Hence,the nonnegative maximum of u in is attained only on on. Now we consider the general case Lu>0 in For any e>0,consider w(z)=u(x)+Eeu=i, where uis a positive constant to be determined.Then, Lw=Lu se"i (anlu2+bu+c). Since b1 and c are bounded and a1 >A>0 in by choosing >0 large enough,we get a1142+b14+c>0in2 This implies Lw>0 in By the special case we just discussed,w attains its nonnegative maximum only on on and hence, xw≤器w+ Then mgxu≤mxw≤rxw+≤器xw+exe We have the desired result by letting e0 and using the fact that on c. 0 If c=0 in we can draw conclusions about the maximum of u rather than its nonnegative maximum.A similar remark holds for the strong max- imum principle. A continuous function in always attains its maximum in D.Theorem 1.1.2aserts that any subo upto the boundary attainsits maximum on the boundary an,but possibly also in Theorem 1.1.2 is 由扫描全能王扫描创建 ▣ 由 扫描全能王 扫描创建