10. The Euler equation for some more general cases.

1. Higher derivatives: Assume that we have a functional of the type

where y  is a function in C2n[a,b] that fulfills the boundary conditions

The Euler equationen is then

2. Several functions: Assume that we have a functional of the type

where yi, i=1,…,n  are functions in C2[a,b] that fulfill the boundary conditions 

<>The Euler equation is then 


3. Several variables: Assume that we have a functional of the type 

where u  is a function in C2( )  that fulfills the boundary condition 

The Euler equation is then

Example 14: (Compare example 9.) The Euler equationen to

is Laplace’ equation

Example 15: (Compare example 8.) The Euler equation to

is the nonlinear differential equation

Example 16: The Euler equation to the functional

is p-Laplace equation

This equation is also called the p-harmonic equation and is sometimes used to model non-Newtonian flows.