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EULER-LAGRANGE equation

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Theorem (EULER-LAGRANGE equation)    [satz:7-4]
Let  E ,  f  as in definition:7-1 with the additional assumption, that  f  is twice continuously differentiable. Then for  u∈C2(clos(Ω);RN) :
∂v E(u)
=
 
∫
Ω
 
∑
k
 vk( f′zk(·,u,∇u)- div (f′pk(·,u,∇u)) ) dLn
+
 
∫
∂Ω
 
∑
k
 vk f′pk(·,u,∇u) •νΩdHn-1 .
This implies:
  • [satz:7-4-(1)] Let  u0∈C1(clos(Ω);RN)  and
    M:= {   u∈C1(clos(Ω);RN)  ;   u = u0 auf ∂Ω  }.
    Then  Tu(M)={  v∈C1(clos(Ω);RN) ; v=0  on  ∂Ω  }  for  u∈M . Moreover,  ∂vE(u)=0  for all  v∈Tu(M) , if and only if the EULER-LAGRANGE equations
    div (f′pk(·,u,∇u)) = f′zk(·,u,∇u)      in Ω for k=1,..., N
    are satisfied, and the boundary condition
    u=u0      on ∂Ω.
  • [satz:7-4-(2)] Let  M=C1(Ω;RN) . Then  Tu(M) = M  for  u∈M . Moreover,  ∂v E(u) = 0  for all  v∈Tu(M)=M , if and only if the EULER-LAGRANGE equations
    div (f′pk(·,u,∇u)) = f′zk(·,u,∇u)      in Ω for k=1,..., N
    are satisfied, and the boundary conditions
    f′pk(·,u,∇u)•νΩ = 0      on ∂Ω for k=1,...,N.

Proof. In satz:7-3 is
∇vk•f′pk(·,u,∇u) = div (vk f′pk(·,u,∇u) - vk   div (f′pk(·,u,∇u)) ,
therefore, by the divergence theorem,
 
∫
Ω
 div (vk f′pk(·,u,∇u)) dLn =
 
∫
∂Ω
 vk f′pk(·,u,∇u)•νΩ dHn-1 .
This proves the representation for  ∂vE(u) .

Now, in satz:7-4-(1) and satz:7-4-(2), choose, for given  k∈{1,...,N} , an arbitrary function  vk∈C01(Ω)  and set  vl=0  for  k≠l . This gives

0 = ∂vE(u) =
 
∫
Ω
 vk(f′zk(·,u,∇u) - div (f′pk(·, u,∇u))) dLn.
Since  vk  is an arbitrary function with compact support, it follows, that the second factor in the integrand has to vanish. This argumentation is a fundamental one in calculus of variations, therefore it is formulated below in in satz:7-5 as a separate lemma. This proves, that the EULER-LAGRANGE equations are satisfied.

To prove satz:7-4-(2), we insert this result into the representation of  ∂vE(u)  and obtain, that for all  v∈C1(clos(Ω);RN) 

0 =
 
∫
∂Ω
 
∑
k
 vk f′pk(·,u,∇u)•νΩ dHn-1 .
Then the same argumentation as above implies the boundary conditions, where satz:7-5 has to be applied to the surface measure  Hn-1  on  ∂Ω  instead to the LEBESGUE measure  Ln  on  Ω .

Notice
The complete characterization of the first variation of  E:M → R  depends on the structure of the set  M , on which  E  is considered. Here we obtain in both cases the EULER-LAGRANGE equation, and in addition in the case
[eq:7-dirichlet]
M={u∈C1(clos(Ω);RN);  u=u0 on ∂Ω}
the DIRICHLET boundary condition
u=u0 on ∂Ω,
and in the case
[eq:7-neumann]
M=C1(clos(Ω);RN)
the NEUMANN boundary condition
f′pk(·,u,∇u)•νΩ = 0     on ∂Ω for k=1,...,N.

The formulation of the following lemma is one possibility among several variants.

Fundamental lemma of calculus of variations    [satz:7-5]
Let  M  be an  m -dimensional  C1 -surface in  Rn ,  1≦m≦n ,  f:M → Rl  continuous. Assume that
 
∫
M
 ζf dHm = 0      for all ζ∈C∞(Rn), for which supp ζ∩M compact in M .
Then  f=0 .

Proof. Assume, there is an  x0∈M  with  f(x0) ≠0 , e. g.  fi(x0)>0  with  i∈{1,...,N } . Then there are  δ0>0  and  c0>0  such that  fi(x)≧c0  for  |x-x0|<δ0 , for  f  is continuous. Choose  ζ∈C∞(Rn)  with  ζ≧0  and  ζ(x0)>0 . For example so, that  ζ(x) ≧c1>0  for  |x-x0|<δ1=δ0/2  and   supp (ζ)⊂Bδ0(x0) . Then
0 = (
 
∫
M
 ζf dHm)•ei =
 
 
∫
M∩Bδ0(x0)


ζfi
≧0
 dHm ≧c0 c1
 
 
∫
M∩Bδ1(x0)
 dHm > 0  ,
a contradiction.
Version 1.7
H.W. Alt - 02.01.2007

First variation First variation
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© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany