1.1.The Maximum Principle 17 Theorem 1.1.12.Letbe a bounded domain in R satisfying an exterior g有Cs。sphere condition at xo∈8 and L be given by(1.l.l）,for some a,bi c()satisfying (1.1.3).Suppose that u is a C()()- solution of Lu=f inn, u=on as, for some feL()nC()andpC(a.Them,for any z∈， |u(x)-u(xo)川≤w(x-xol), wherewis anondecreasing continuous function inD),with D-diam() and limw(r)=0,depending only on n,A,A,the Lo-norms of bi and c diam(),R in the exterior sphere condition,supmaxsup and the modulus of continuity ofp on 8. Proof.Set L0=a0j+b10, Then,Lou=f-cu in 9.Let w=wo be the function in Lemma 1.1.11 for Lo,i.e., Low≤-1in2, and,for any zeon\{ro}, w(xo）=0,w(x)>0. We set F=sgPf-c4,重=mxol. Then, Lo(±u)≥-Fin2. Let e be an arbitrary positive constant.By the continuity of at o,there exists a positive constant 6 such that,for any eBs(o), lp(x)-p(xol≤e. We then choose K sufficiently large so that K>F and Kw≥2Φon8n\B(xo: We point out that K depends on e through the positive lower bound of w on as\Bs(xo).Then, Lo(Kw)≤-Fin2, 1+k3F可. and lp-p(ro训≤e+Kw on s.为hn4外专衣.· 由扫描全能王扫描创建 ▣ 由 扫描全能王 扫描创建