e 1.Linear Elliptic Equntions In view of (ro)=0,v attains at ro its maximum in D.Hence,we obtain mnl≥. and then 0w2- ()=2cpRe-e>0. This is the desired result. ◇ Theorem 1.1.6still holds if we for Bany bounded domain which satisfies an interior sphere condition at roea,namely,if there exists a ball B C with ro E B.This is because such a ball B is tangent to on at ro.We note that the interior sphere condition always holds for C2-domains. Now,we are ready to prove the strong maximum principle due to Hopf 8 Theorem 1.1.7.Let be a bounded domain in R"and L be given by (1.1.1),for some bcC(S)satisfvingein and (11.2).Suppose that u E C()C2()satisfies Lu 20 in S.Then,u attains only on 89 its nonnegative marimum in S unless u is constant. Proof.Let M be the nonnegative maximum of u in and set D={x∈2：u(x)=M}. We prove either D=0or D=by contradiction.Suppose D is a nonempty relat: ve such that oBnD0.In fact,we may choose a point z.\D with dist(.D)<dist(.,0)and then take the ball centered at z.with the radius dist(.,D).Suppose.Obviously,we have Lu≥0inB and u(x)<u(xo)for any x∈B and u(xro）=M≥0. By Theorem 1.1.6,we have 0ew>0 where v is the exterior unit normal to B at to.On the other hand,xo is an interior maximum point of u in This implies Vu(xo)=0,which leads to a contradiction.Therefore,either D=0 or D=.In the first case,u attains only on on its nonnegative maximum in while in the second case, u is constant in s. □ 由扫描全能王扫描创建由 扫描全能王 扫描创建