Chapter 6 Partial Differential Equations

6.1. INTRODUCTION 3
In this case we define the differential operator L = ∑ |α|≤k a α(x)∂ α and write Lu = f. More
generally, we have the quasi-linear equations which have the form
∑
a α (x, (∂ β u) |β|≤(k−1) )∂ α u = b(x, (∂ β u) |β|≤(k−1) ). (6.1.4)
|α|≤k
Thus the partial differential equation is linear if u and all of its derivatives appear in a
linear fashion. For example, the general form of a linear 2 nd order partial differential equation
is
Au xx + Bu xy + Cu yy + Du x + Eu y + Fu = G
For instance,
yu xx + u yy + u x =0
is linear and
uu xx + u yy + u x =0
is nonlinear but quasi-linear. If G = 0 the equation is homogeneous.
Some general concerns in the study of partial differential equation’s are:
1. existence of solutions,
2. uniqueness of solutions,
3. dependence of solutions on the ‘data’,
4. smoothness, or regularity of the solution, and
5. representations for solutions and behavior of solutions.
The first three of these issues are related to the notion of a well-posed problem. In
particular, a problem is well posed in the sense of Hadamard if there exists a unique solution
that depends continuously on the data.
Consider the following examples in R 2 ,
1. The general solution of ∂ 1 u =0isu(x 1 ,x 2 )=f(x 2 ) where f is arbitrary. Thus the
equations gives complete information about the behavior of the solution with respect
to x 1 (it is a constant) but it gives no information with respect to the x 2 variable.
2. The general solution of ∂ 1 ∂ 2 u =0isu(x 1 ,x 2 )=f(x 1 )+g(x 2 ) where f and g are
arbitrary. Thus, other that the fact that the dependences on x 1 and x 2 are “uncoupled”
we learn nothing about the dependence on the variables.
3. Any complex valued solution u of the “Cauchy-Riemann” equation ∂ 1 u + i∂ 2 u = 0 is a
holomorphic function of z = x 1 + ix 2 . Thus they are in particular C ∞ . The equation
imposes very strong conditions on all derivatives of the solutions.