Partial Differential Equations/Calculus of variations

Partial Differential Equations
 ← Sobolev spaces Calculus of variations Bochner's integral → 

Calculus of variations is a method for proving existence and uniqueness results for certain equations; in particular, it can be applied to some partial differential equations. The method works as follows: Let's say we have an equation which is to be solved for the variable (this variable can also be a function). We look for a function whose minimizers satisfy the equation, and then prove that there exists a minimizer. We have thus obtained an existence result.

In some cases, we will additionally be able to show that values satisfying the equation are minimizers of the function. If we now find out about the number of minimizers of the function, we will also know the numbers of solutions to the equation. If then the function has only one minimizer, we have obtained a uniqueness result.

Sometimes, calculus of variations also works ‘the other way round’: We have a function whose minimizers are difficult to find. Then we show that the minimizers of this function are exactly the solutions of a partial differential equation, which is easy to solve. We then solve the partial differential equation in order to obtain the minimizers of the function.

Strong convexityEdit

"Normal" equationsEdit

Consider the equation system

 

for functions  . If there exists a function   such that

 

we find that the equation system   is satisfied if and only if

 

If   satisfies the right conditions, we have   at exactly one point  :

Definition 13.1:

Let  , and let's denote the Hessian matrix of   at   by  .   is called strongly convex iff

 

Theorem 13.2:

Let   be strongly convex. Then   has exactly one critical point (i. e. a point   where  ).

Proof:

From   being strongly convex it follows that for all  ,   is positively definite. Therefore, every critical point is a local minimum (this is due to the sufficient condition for local minima). Thus, it suffices to prove that there is exactly one local minimum.

1.

We show that there exists a local minimum.

We take Taylor's formula around  :

 

Thus,

 

for a  . Therefore, there exists an   such that

 

By the extreme value theorem, there exists a minimum   of   in  . It can not be attained on the border, because if  , then   and thus by    , which would imply that   is not a minimum. Therefore it is attained in the interior and is thus a local minimum. In fact, from   and from   being a minimum on   even follows that it is a global minimum of  .

2.

We show that there is only one local minimum.

Let   and   be two local minima. We show that  , thereby excluding the possibility of two different minima. We define a function   as follows:

 

Let's calculate the first and second derivative of  :

 

Since   and   are local minima,   and  . Therefore,

 

and

 

Therefore, by the mean value theorem, there exists a   such that

 

But since

 

,   implies  . 

Corollary 13.3:

Suppose we have an equation system

 

If there is a function   which is strongly convex and

 

, then the equation system   has exactly one solution.

Proof: See exercise 1.

Example 13.4:

Another example is given in exercise 2.

Elliptic partial differential equationsEdit

Euler-Lagrange equationsEdit

The ‘brachistochrone problem’Edit

ExercisesEdit

SourcesEdit

Partial Differential Equations
 ← Sobolev spaces Calculus of variations Bochner's integral →