Uniformly Elliptic PDEs with Bounded, Measurable Coefficients
Uniformly Elliptic PDEs with Bounded, Measurable Coefficients
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具有有界、可测系数的均匀椭圆偏微分方程
DOI：
10.1007/s00041-001-4031-6
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发表时间：
1995
影响因子：
1.2
通讯作者：
R. Jensen
中科院分区：
文献类型：
--
作者：
R. Jensen
设是一个定义在定义域上的非发散线性二阶一致椭圆型偏微分算子，考虑函数u何时是函数的解？“天真的答案，‘u是所有人的onifandfor all的解决方案’显然太有限了。 事实上，如果L的系数是，那么L可以被重写为发散形式，其中可以应用“弱”解的概念。 在这种情况下，可能有无穷多个函数是“弱”的，但不是经典解。 更重要的是，即使L的系数是有界的和可测的，Krylov最近的结果允许我们构造的“解决方案”，这些“解决方案”一般不优于连续的，“弱”的解决方案，前面提到的可以得到这种建设，太。前面的讨论为我们提供了一个充分的解的外在定义（即，给定一个函数u，我们要么证明它是或不是这样一个构造的结果），这个构造已经被几个作者使用过，但是不是特别令人满意或有启发性的。 我们在本文中的主要贡献是显示以下内容。I.有一个内在的定义的解决方案是等价的外在的。二.此外，内禀定义只是（现在）众所周知的克兰德尔-狮子粘度解决方案，以自然的方式修改，以适应可测量的系数。
Letbe a nondivergent linear second-order uniformly elliptic partial differential operator defined on functions with domainConsider the question, "When is a function u a solution ofon?" The naive answer, "u is a solution ofonifandfor all" is clearly too limited. Indeed, if the coefficients of L are inthen L can be rewritten in divergence form for which the notion of a "weak" solution can be applied. In this case there could be infinitely many functions that are "weak" but not classical solutions. More importantly, even if the coefficients of L are just bounded and measurable, the recent results of Krylov permit us to construct "solutions" ofonand these "solutions" are generally no better than continuous; the "weak" solutions previously mentioned can be obtained by this construction, too. The preceding discussion provides us with an adequate extrinsic definition of solution (i.e., given a function u we either prove that it is or is not the result of such a construction) that has been used by several authors, but one that is not particularly satisfying or illuminating. Our major contribution in this paper is to show the following. I. There is an intrinsic definition of solution that is equivalent to the extrinsic one. II. Furthermore, the intrinsic definition is just the (now) well-known Crandall-Lions viscosity solution, modified in a natural way to accommodate measurable coefficients.