用户: Yao/现代偏微分方程课程重点
1函数的正则化
Take a fixed supported in the unit ball with . Writethen is supported in , with .
Proposition 1.1. For where , we have
(1) | , where , |
(2) | , |
(3) | as . |
Proof.
(1) | The smoothness of is derived from interchanging integration and derivatives. |
(2) | By Minkowski, . |
(3) | First show that holds for , then use to extend the result to the case. |
Exercise. Show that a.e. as .
2广义函数
Definition 2.1. A distribution is a continuous linear functional on the space of test functions like . The convergence of distributions is defined to be the weak convergence of functionals.
The construction of functions is called the regularization of distributions.
Proposition 2.2.
(1) | A linear functional on is continuous iff for any compact there exists such that for all smooth functions supported in . |
(2) | A linear functional on is continuous iff there exists compact and such that for all smooth functions . |
Definition 2.3 (Fourier transform).
Proposition 2.4 (Parseval’s relation).
Definition 2.5 (Calculation on distributions).
(1) | . |
(2) | . |
(3) | . |
(4) | . |
(5) | . |
Proposition 2.6.
(1) | , |
(2) | , |
(3) | . |
Exercise. Suppose that equals to in the first quadrant, and in the rest three. Find .
Exercise. Suppose that is the characteristic function of the unit disc . Find .
3Sobolev 空间 (上)
Definition 3.1. () is the Banach space of functions with the norm
Definition 3.2.
Remark 3.3. while for , since convergence implies uniform convergence on , and consequently a sequence of functions with compact support converges to a function that also tends to at .
Exercise. Show that is not dense in for .
Definition 3.4. For , is the Hilbert space of functions with the norm
Remark 3.5. We may write , and the expression becomes , which is easier to memorize.
Proposition 3.6 (Trace Theorem). For , , belongs to , and is a bounded operator.
Proposition 3.7 (Lions Extension). For a bounded region with boundary, can be extended to (which means equals in ) in such a way that .
Proof. A partition of unity reduces the question to the case of is compactly supported, and flattening out the the boundary by a coordinate change to the case of where .
Assume that first. For arbitrary distinct numbers , let
Let where the coefficient matrix is a nonsingular Vandermonde matrix, so the solution exists. Then and is continuous for on and , so .
4Sobolev 空间 (下)
Proposition 4.1.
Proof. The first is becuase
For the second and the third, see Sobolev inequalities.
The fourth is becuase
For the fifth, by the second step of Theorem 1.4 of Sobolev inequalities, for appropriate ,
Proposition 4.2 (Rellich–Kondrachov). For a bounded region with smooth boundaries, the inclusion is compact for .
Proof. Enough to deal with , since the composite of compact operators is compact.
For each bounded sequence in , which is also bounded in , write . First, we have uniformly as because and for , and hence for ,where is a constant dependent on .
Next, we observe that for fixed , is compact for it is uniformly bounded and equicontinuous as and .
Proposition 4.3 (Poincaré). is a domain lying between two parallel hyperplanes of distance . For , .
5椭圆方程
Let be a second order elliptic partial differential operator where the matrix for all .
For , acts on in the sense of distributions, and hence .
Proposition 5.1 (Gårding). For , .
Proposition 5.2. There is a number such that for each , there exists a unique weak solution of for each .
Proof. Introduce the formal adjoint operator defined as (one easily checks ) Take to be of the preceding proposition. Then for we have which is
6Galerkin 方法
Solve the equation
Let be the set of eigenfunctions of , which serves as an orthonormal basis of both and . (See problem 5.3 of this page.)
We look for such that vanishes on (that is, its orthognal projection onto this subspace is zero), and approximates in certain sense.
First, we carry out the calculation of .
For , satisfiesthen integration by parts yieldswhich are linear equations of .
7能量估计
1. For the energy estimate for hyperbolic equations, see the last problem of this page.
2. We now establish the energy estimate for the parabolic equationwhere .
We have
Let , then . Sincewe get that , and by Gronwall we conclude that
3. Consider the symmetric hyperbolic systemwhere , , are symmetric matrices, , and is a matrix .
WLOG, assume that , for we can replace by for big enough if otherwise. And assume that is bounded.
Multiply both sides by , we get
sowhere due to Green’s formula and some boundary conditions. Then by Gronwall,
8算子半群
is a Banach space.
Definition 8.1. is called a semigroup if , , and for all the map is continuous, and is called a contraction semigroup if in addition for all .
Definition 8.2. is called the domain of .
is called the (infinitesimal) generator of the semigroup.
Theorem 8.3 (Hille–Yosida). For a densely defined closed linear operator in , the following assertations are equivalent:
(1) generates a -semigroup of contractions on ;
(2) s.t. , is invertible and ;
(3) s.t. , is invertible and .
The textbook of S.X.Chen establishes the equivalence of the first two assertations, while what Y. Zhou teaches is the equivalence of the first and the third.
Define for . Then for ,as .
Let . Since , we have . Sinceas for and hence also for , we can define . Since is indeed the generator of .
参考文献
[Evans] | L. C. Evans. Partial Differential Equations. Graduate Studies in Mathematics 19. Amer. Math. Soc, 2010. |
[Le] | G. Leoni. A First Course in Sobolev Spaces. Graduate Studies in Mathematics 181. Amer. Math. Soc, 2017. |