On the unique continuation theorem for certain second and fourth order elliptic equations
On the unique continuation theorem for certain second and fourth order elliptic equations
复制标题
关于某些二阶和四阶椭圆方程的唯一连续定理
DOI：
10.1002/cpa.3160110104
复制
发表时间：
1958
影响因子：
3
通讯作者：
R. Pederson
中科院分区：
文献类型：
--
作者：
R. Pederson
实解析函数的性质是，如果两个函数的差在某一点足够快地消失，则它们在它们的共同定义范围内是相同的。这就是解析函数的唯一连续性质。自然会出现这样一个问题：还有哪些函数类满足唯一的连续定理？一些椭圆型偏微分方程解已被证明具有这一性质，但沿这条线的信息非常稀少。本文的目的是在这个方向略微扩展已知的信息。定义在欧氏空间D中的实偏微分算子称为椭圆算子，如果齐次m次多项式n$(z，6)=2 all“‘qf，**5 cm il…由最高次系数构成的Im=1对每个x E D，6#0满足条件p(x，6)>；0。这个条件意味着算子A是偶数阶的，因为齐性$(x，6)=(-L)m$(z，6)。Carleman[3]证明了具有连续可微引导系数的二维二阶椭圆型方程的唯一延拓定理。Morrey[13]和Bers-Nirenberg[12]证明了如果主导系数仅仅是Holder连续的，则Carleman定理是正确的。约翰[S]证明了如果A的系数是解析的，则AU=0的任何解G也是解析的，因此具有唯一的延拓性。最近，几位作者证明了二阶椭圆型偏微分方程解(不一定具有解析系数)解的唯一延拓定理。
Real analytic functions have the property that if the difference of two vanishes sufficiently fast at a point then they are identical in their common range of definition. This is known as the unique continuation property of analytic functions. The question naturally arises which other classes of functions satisfy the unique continuation theorem? Solutions of certain partial differential equations of elliptic type have been shown to have this property, but information along this line is very meager. It is the purpose of this paper to extend slightly the known information in this direction. The real partial differential operator defined in a domain D in Euclidean%-space is said to be elliptic if the homogeneous m-th degree polynomial n$(z, 6)= 2 ail"'qf,** 5Cm iL... im= 1 formed with the coefficients of highest degree satisfies the condition p (x, 6)> 0 for every x E D, 6# 0. This condition implies that the operator A be of even order, for, by homogeneity $(x, 6)=(-l) m $(z, 6). Carleman [3] proved the unique continuation theorem for second order elliptic equations in two dimensions with continuously differentiable leading coefficients. Morrey [13] and Bers-Nirenberg [12] showed that Carleman's theorem is true if the leading coefficients are merely Holder continuous. F. John [S] has shown that if the coefficients of A are analytic, any solution G of AU= 0 is also analytic and hence has the unique continuation property. Several authors have recently proved unique continuation theorems for solutions of the second order elliptic partial differential equation (with not necessarily analytic coefficients)