Bibliography
[1] Wu Zhuoqun et al. Introduction to elliptical and parabolic equations. Beijing: Science Press, 2003
Anisotropic Sobolev space and Holder space
Since the "status" of space variables and time variables in parabolic equations are different, the function space used when studying parabolic equations is somewhat different from the function space used when studying elliptic equations.
The convention represents the gradient of the function with respect to the spatial variable, and the weak derivative of the function with respect to time is expressed as.
Let the anisotropic Sobolev space be an open set in , note.
Definition 1 Assuming it is a non-negative integer, the set is said to be assigned a norm
The resulting normed linear space is a Sobolev space. This is a Banach space in which the order of the weak derivatives of the elements is lower than the order of the weak derivatives.
Definition 2 is set to or, . It is said that the set is assigned a norm
The resulting normed linear space is Sobolev space.
At that time, the inner product could be defined in the above function space to become a Hilbert space.
At that time, there was.
Definition 3: Remember sum.
(1) Use the set consisting of functions that are infinitely differentiable in the vicinity and on.
(2) A set composed of functions that are infinitely differentiable in the vicinity.
(3) Use closures that represent in. Use closures that represent within. Use closures that represent within. Use closures that represent within.
Definition 4: assign norm
The resulting Banach space.
Definition 5 Note, where the inner product is defined as anisotropic Holder space
Parabolic distance: Following the previous notation, for any two points, define the parabolic distance between them as
Definition 6 Suppose is the function defined on, for, introduce Holder semi-norm
Use the expression to satisfy all functions, and define the norm as
For non-negative integers, define the function space
and define the semi-norm
and norm
This is a Banach space, and at that time, there
where and represents the sum of the norms taken for all second-order derivatives with respect to the spatial variable.
Anisotropic embedding theorem
Theorem (embedding) Suppose it is a bounded region, , then
Theorem (embedded) Note, then suppose, there is
Theorem (Anisotropic Poincare Inequality) Suppose, .(1) If, then (2) If, then
Machine-translated from the Chinese original 现代偏微分方程补充笔记(时间变量向异性Sobolev空间与Holder空间) (published 2024-03-27). WeChat original ↗