Chapter 6 Partial Differential Equations

2 CHAPTER 6. PARTIAL DIFFERENTIAL EQUATIONS
A partial differential equation of order k is an equation of the form
F (x 1 ,x 2 , ··· ,x n ,u,∂ 1 u,...,∂ n u, ∂ 2 1u, ··· ,∂ k nu) = 0 (6.1.1)
relating a function u of x =(x 1 , ··· ,x n ) ∈ R n and its partial derivatives of order ≤ k.
Given numbers a α with |α| ≤k, we denote by (a α ) |α|≤k . the element in R N(k) given by
ordering the α’s in any fashion, where N(k) is the cardinality of {α : |α| ≤k}. Similarly, if
S ⊂{α : |α| ≤k} we can consider the ordered (card S)-tuple (a α ) α∈S .
NowletΩbeanopensetinR n , and let F be a function of the variables x ∈ Ω and
(u α ) |α|≤k . Then we can write the partial differential equatio of order k as
F (x, (u α ) |α|≤k )=0. (6.1.2)
A function u is called a classical solution of this equation if ∂ α u exists for each α in F , and
F (x, (u α (x)) |α|≤k )=0, for every x ∈ Ω.
We denote by C(Ω) the space of continuous functions on Ω. If Ω is open and k is a
positive integer, C k (Ω) will denote the space of functions possessing continuous derivatives
up to order k on Ω, and C k (Ω) will denote the space of all u ∈ C k (Ω) such that ∂ α u extends
continuously to the closure of Ω denoted by Ω for all 0 ≤|α| ≤k. We also define
C ∞ (Ω) = ∩ ∞ k=1C k (Ω),
C ∞ (Ω) = ∩ ∞ k=1C k (Ω).
If Ω ⊂ R n is open, a function u ∈ C ∞ (Ω) is said to analytic in Ω if it can be expanded
in a convergent power series about every point of Ω. That is, u is analytic in Ω if for each
x ∈ Ω there exist an r>0 so that for all y ∈ B r (x) ={y : |y − x|