Well-Posed Problems for a Partial Differential Equation of Order $2m + 1$
Well-Posed Problems for a Partial Differential Equation of Order $2m + 1$
复制标题
$2m 1$ 阶偏微分方程的适定问题
DOI：
10.1137/0501020
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发表时间：
1970
影响因子：
2
通讯作者：
R. Showalter
中科院分区：
文献类型：
--
作者：
R. Showalter
包含2m阶椭圆微分算子M和<；2m阶微分算子L。利用Hilbert空间方法建立和求解该问题的抽象形式，并讨论解的存在唯一性、渐近性和边界条件。广义问题的表述是1的目标，我们有理由考虑两种类型的解，称为弱和强。给出了算子M上广义问题弱解存在唯一性的充分条件。这些条件构成了M上的椭圆假设，并在3中作了简要讨论。L的类似假设导致了弱解的渐近行为的结果。文2讨论了M和L相等且自伴的情况，也就是在这里，方程的系数7的作用首先出现。尽管它很特殊，但这是一种在应用中经常出现的情况，人们对这个系数7[4]，[25]很感兴趣。弱解和强解不仅通过正则性条件来区分，还通过它们的相关边界条件来区分。它首先出现在5中，可以在强解上规定太多(独立的)边界条件，但在应用程序中可以看到，这些条件的相互依赖性被建立在关于算子的域的假设中。1.广义问题。设G是n维实欧几里德空间R“中的一个非空开集，它的边界G是一个n维流形，G位于它的一侧.C(G)是G上的无限可微函数空间，C(G)是C(G)的线性子空间，C(G)是G中具有紧支撑的函数组成的线性子空间.Sobolev空间Hm(G)=H是L2(G)中函数(等价类)的Hilbert空间，其通过m阶的分布导数都属于L2(G).给出了内积和范数.
containing the elliptic differential operator M of order 2m and the differential operator L of order <__ 2m. Hilbert space methods are used to formulate and solve an abstract form of the problem and to discuss existence, uniqueness, asymptotic behavior and boundary conditions of a solution. The formulation of a generalized problem is the objective of 1, and we shall have reason to consider two types of solutions, called weak and strong. Sufficient conditions on the operatorM are given for the existence and uniqueness ofa weak solution to the generalized problem. These conditions constitute elliptic hypotheses onM and are discussed briefly in 3. Similar assumptions on L lead to results on the asymptotic behavior of a weak solution. The case in whichM and L are equal and self-adjoint is discussed in 2, and it is here that the role of the coefficient 7 of the equation appears first. Special as it is, this is a situation that often arises in applications, and there has been considerable interest in this coefficient 7 [4], [25]. The weak and strong solutions are distinguished not only by regularity conditions but also by their associated boundary conditions. It first appears in 5 that it is possible to prescribe too many (independent) boundary conditions on a strong solution, but in the applications it is seen that the interdependence of these conditions is built into the assumptions on the domains ofthe operators.Two examples of applications appear in 6 with a discussion of the types of boundary conditions that are appropriate. 1. The generalized problem. LetG be anonempty open set in the n-dimensional real Euclidean space, R", whose boundaryG is an (n 1)-dimensional manifold with G lying on one side of it. C(G) is the space ofinfinitely differentiable functions on G, and C(G) is the linear subspace of C(G) consisting of functions with compact support in G. The Sobolev space Hm(G)= H is the Hilbert space of (equivalence classes of) functions in L2(G), all of whose distributional derivatives through order m belong to L2(G). The inner product and norm are given,