Bibliography
[1] Chen Shuxing. Introduction to Modern Partial Differential Equations (Second Edition). Beijing: Science Press, 2018.
Generalized solution to the Dirichlet problem
Assuming that is a bounded region with smooth boundaries, discuss the boundary value problem of the elliptic equation of the following form
Here, the coefficient satisfies the symmetry condition and the following elliptical condition
where is a positive constant that has nothing to do with, the condition is called the homogeneous Dirichlet condition, and the boundary value problem is calledDirichlet boundary value problem.
The solution in classical theory must be found in . Next, we need to find a suitable generalized function space to reduce the above requirements.
Example (film balance problem)Examining the membrane equilibrium problem with a fixed boundary on a planar region, it can be reduced to finding the energy functional
The problem of finding the minimum value in the function class that takes zero value on the boundary is equivalent to the solution of the homogeneous Dirichlet problem of the Possion equation. In fact, assuming that the minimum value is the element, then for any,
Thereby there is
Since only the function and the square of its first-order partial derivative appear in the expression of , and it can be seen from the example that if the minimum value sought is limited to a function class with a boundary value of zero, it may not be possible to find an element that makes the functional reach a minimum. Therefore, the range of variation of the function is expanded, and the extreme value is considered in the space of modular completion of the function class with a boundary value of zero.
Note that at that time, the integral by parts was the derivation operation of a generalized function. Therefore, if it is an element that takes a minimum value, it must be satisfied according to the meaning of a generalized function, and it can be known from the trace theorem that the trace on is zero.
At that time, , so the formulation of the original problem was changed to: for , seek such that it satisfies according to the generalized function. This solution is called the generalized solution.
Regularity problem: If it has high regularity, what kind of regularity will the solution have? Specifically, if when , what kind of space does it belong to? According to the embedding theorem, if and when , it has a second-order continuous partial derivative, which is a classical solution.
The boundary condition can also be non-homogeneous, that is, here is a function given above. The processing method is to convert it into a homogeneous boundary condition. If it is defined above, it can be extended to a function, still in memory, so that it will satisfy
Generalized solution of the second (third) boundary value problem
Now we discuss the Neumann problem of the equation. At this time, the boundary conditions are modified to
If , then we get it from Green’s formula
Here is a bilinear form of
Therefore, when satisfied, there is
For the third boundary value problem, consider the boundary conditions
At that time, it satisfies the boundary conditions according to the meaning of the trace.
Machine-translated from the Chinese original 现代偏微分方程笔记(椭圆型方程边值问题的广义解) (published 2024-02-22). WeChat original ↗