10 1.Linear Elliptic Equations referred to as the weak maximum principle.A stronger version asserts that subsolutions attain their maximum only on the boundary,unless they are constant. As a simple consequence of Theorem 1.1.2,we have the following result Corollary 1.1.3.Let be a bounded domain in Rm and L be given by （1.1,for some a,b,c∈L()nC(2)satisfying c≤0nand (1.l.2).Suppose that u∈C()nC2()satisfies Lu≥0 in and u≤0on an.Then,u≤0inn. More generally,we have the following comparison principle. Corollary 1.1.4.Let be a bounded domain in R and L be given by (1.l.1),for some aij,b,c∈L(2)nC(）satisfying c≤0 in n and (1.1.2).Suppose that u,v∈C(①)nC2()satisfy Lu≥Lv in and u<u onan.Then,u≤vinn. The comparison principle provides a reason that functions u satisfying Lu≥fare alled subsolutions.They are less than a solution v of Lv=f with the same boundary value. In the following,we simply say by the maximum principle when we apply Theorem 1.1.2,Corollary 1.1.3,or Corollary 1.1.4. A consequence of the maximum principle is the uniqueness of solutions of Dirichlet problems. Corollary 1.1.5.Let S be a bounded dom ain in Rn and L be given by (1.1.1),for some a4j,bi,c∈L∞()nC(）satisfying c≤0 in and (1.1.2).Then for any fC()and C(8s),there exists at most one solution uEC()nC2(S)of Lu=f in, u=on 8n. 1.1.2.The Strong Maximum Principle.The weak maximum principle asserts that subsolutions of linear elliptic equations attain their nonnegative maximum on the boundary under suitable conditions.In fact,these subso- lutions can attain their nonnegative maximum only on the boundary,unless they are constant.This is the strong maximum principle. For any Cl-function u in that attains its maximum on 80,say at zoas,we have(o)20,where v is the exterior unit normal to at zo.The Hopf lemma asserts that this normal derivative is in fact positive if u is a subsolution in Theore m 1.1.6.Let B be an open ball in R"with o EB and L be given by (1.1.1),for some aij,bi,c L(B)nC(B)satisfying c<0 in B and 由扫描全能王扫描创建由 扫描全能王 扫描创建