Bibliography
[1] Chen Shuxing. Introduction to Modern Partial Differential Equations (Second Edition). Beijing: Science Press, 2018.
A generalized function is a continuous linear functional on the basic function space.The linear continuous functional on is called a generalized function, the linear continuous functional on is called a generalized function, and the linear continuous functional on is called a generalized function. All these three types of generalized functions respectively form the corresponding generalized function space, which is recorded as the relationship expression with.
Normal functions (Lebesgue integrable functions) can correspond to a specific generalized function. Generally, this normal function and its corresponding generalized function in are regarded as the same.
Example 1 Any locally integrable function is a generalized function.
In fact, assuming that it is a locally integrable function, if
Example 2 The function is a generalized function.
Theorem 1 If , then for any compact set, there are constants and positive integers, such that
This is true for any function that satisfies the conditions. Vice versa.
Theorem 2 If , then there are compact sets and constants such that for any function, vice versa.
Definition 1The coset of the largest open set that makes a generalized function take zero value is called the generalized functionsupport, recorded as.
For generalized functions, generally we cannot say the value of a generalized function at a specific point, but its value in any open set has exact meaning.
If there is a generalized function, for any function, where, holds
Theorem 3If there is a generalized function, for any point, there is a neighborhood such that it is zero in it, then there is .
Theorem 4If there is an open set group covering the compact set in , we can find the following function group, where and in .
Example 3 The support of the function is the origin.
Theorem 5For any generalized function, its supports are compact. On the contrary, for any generalized function, if its supports are compact, it must be a generalized function.
Definition 2If there is a generalized function sequence belonging to a certain basic function space, for any element in its basic function space, it is true
weak convergence
Example 4 Suppose it is a function sequence defined as follows: Using the mean value theorem of integrals, we have Example 5 Suppose, then we have.
In fact, for any function, using the Riemann-Lebesgue lemma, we know
Theorem 6 If , then it is a function of .
proveNote that for a fixed , there are , so it is meaningful. At that time, the supports that can be made all fall into a fixed compact set (such as a closed sphere with the center as the radius), and the derivatives of each order in all converge to the derivatives of each order. Therefore, at that time, there is
Recorded as the increment of the direction, construct the difference quotient
Starting from a generalized function, the function obtained is called the regularization of the generalized function. Theorem 7: When, there is. Theorem 8: When, there is.
Definition 3 Let it be a generalized function, and define the partial derivative with respect to be a generalized function, satisfying
It can be known from the topology of , and can be obtained from , so the above formula determines a linear continuous functional on , which is also a generalized function, called aboutgeneralized derivative.
For heavy indicators, there are
We obtain the following two properties of generalized functions:
(1) Any order derivative of a generalized function exists.
(2) The derivative of a generalized function has nothing to do with the order of its derivation.
Example 6 Consider the Heaviside function
As a generalized function, it is differentiable on, for any
Example 7 is denoted as the Cauchy principal value of the divergent integral, that is
Therefore, it is the derivative of . And it is a locally integrable function, with, therefore.
Theorem 9 If the derivative of a generalized function in a connected open set is zero, then it is always a constant in .
Theorem 10If a generalized function has a support of , then it must be a finite linear combination of the function and its derivatives:
Here is a certain positive integer, is a constant coefficient, and this expression is unique.
Example 8 Suppose, find the solution of the generalized function equation.
Solution Consider the solution first. The support known from this formula is at the origin, and it can be seen from Theorem 10
And for all, so, where is any constant. Then it can be seen that the solution of the equation is. It will be written as, since
Let it be a generalized function, , and define the product of
multiplier
Example 9 is not a multiplier of .
In fact, it is an element of , but no longer belongs to .
If it is a generalized function and is the corresponding multiplier, the Leibniz formula is established
Variable substitution If, then it is true for the normal function, and its convolution is defined as
Definition 4 is assumed to be two generalized functions, then the convolution is defined as
Not any two generalized functions can be convolved, but we have
Theorem 11If it is a generalized function, it is a generalized function, then it is a generalized function. And if it is a generalized function, it is a generalized function, the same is true.
The generalized function as a whole constitutes a commutative algebra with identity elements regarding ordinary linear operations and convolution operations. The generalized function is its identity element. This algebra is calledconvolution algebra.
Theorem 12 Suppose, and at least two of them have compact supports, then (1) .(2) .(3) .(4) .
(5) For any heavy index, if at least one of the generalized functions has a compact support set, then
According to this property, if there is a partial differential operator with constant coefficient acting on the convolution of two generalized functions, then we have
(6) If , then it is meaningful, and it is established at this time
Machine-translated from the Chinese original 现代偏微分方程笔记(广义函数及其运算) (published 2024-01-13). WeChat original ↗