REMARKS ON NONLINEAR UNIFORMLY PARABOLIC EQUATIONS
REMARKS ON NONLINEAR UNIFORMLY PARABOLIC EQUATIONS
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关于非线性一致抛物线方程的注解
DOI：
10.1512/iumj.1998.47.1561
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发表时间：
1998
影响因子：
1.1
通讯作者：
A. Swiech
中科院分区：
文献类型：
--
作者：
M. Crandall;K. Fok;M. Kocan;A. Swiech
本文为讨论完全非线性抛物型方程提供了一些工具。其中包括：证明了极大值原理对粘性解在Ln +1意义下的强”解的L1估计仍然成立，对在Ln +1中具有强迫项的极值方程的解在Lp中的梯度估计（p <（n + 1）（n + 2）），利用这一估计改进了极大值原理所提到的p的取值范围（得到了某些p < n+1-但不含接触集），证明了当p < n + 1时，在Lp中具有强迫项的极值方程Dirichlet问题的强可解性，以及二次抛物可导性a.e.对于（n + 2）=2 <p.0，W2;1;p函数的导论.在这项工作中，我们提供了一些工具，在适当的结构条件下的非线性抛物方程的讨论。特别是，我们将在目前的文献中的差距，从而证明了一个完整的推广的“最大值原理”（见下文）的粘性解的某些极端的完全非线性方程可测量的强迫项。在进行这项任务时，我们必须解决某些存在问题，这也是这样做的。所得到的结果制定的标准极值方程，使他们适用于许多其他方程。此外，还对已知结果（不同程度上）给出了新的证明.设0 <为常数，P <0 <1，(X) = ? trace（X +）+ trace（X？）对于X 2 S（n），表示真实的对称n n矩阵的集合.这里trace（X +）（分别为trace（X？））是X的正特征值之和（分别，？X）。设0是另一个常数。讨论了抛物型不等式解的几个结果
This paper provides a number of working tools for the discussion of fully nonlinear parabolic equations. These include: a proof that the maximum principle which provides L 1 estimates of \strong" solutions of extremal equations by L n+1 norms of the forcing term over the \contact set" remains valid for viscosity solutions in an L n+1 sense, a gradient estimate in L p for p < (n + 1)(n + 2) for solutions of extremal equations with forcing terms in L n+1 , the use of this estimate in improving the range of p for which the maximum principle rst alluded to holds (obtaining some p < n+1-but without the contact set), a proof of the strong solvability of Dirichlet problems for extremal equations with forcing terms in L p for some p < n + 1, and the twice parabolic diierentiability a.e. of W 2;1;p functions for (n + 2)=2 < p. 0. Introduction. In this work we provide a number of tools for the discussion of nonlinear parabolic equations under appropriate structure conditions. In particular, we ll gaps in the current literature and thereby prove a full generalization of the \maximum principle" (see below) to viscosity solutions of certain extremal fully nonlinear equations with measurable forcing terms. While going about this task, we must resolve certain existence questions, and this is done as well. The results obtained are formulated in terms of standard extremal equations so that they apply to many other equations. In addition, some new proofs of known (to varying degrees) results are given. Let 0 < be constants and deene P ? (X) = ? trace(X +) + trace(X ?) for X 2 S(n), the set of real symmetric n n matrices. Here trace(X +) (respectively, trace(X ?)) is the sum of the positive eigenvalues of X (respectively, ?X). Let 0 be another constant. We discuss several results concerning solutions of the parabolic inequality