Direct method in the calculus of variations

The method

The calculus of variations deals with functionals J : V → R ¯ , where V is some function space and R ¯ = R ∪ { ∞ } . The main interest of the subject is to find minimizers for such functionals, that is, functions v ∈ V such that: J ( v ) ≤ J ( u ) ∀ u ∈ V .

The standard tool for obtaining necessary conditions for a function to be a minimizer is the Euler–Lagrange equation. But seeking a minimizer amongst functions satisfying these may lead to false conclusions if the existence of a minimizer is not established beforehand.

The functional J must be bounded from below to have a minimizer. This means

inf { J ( u ) | u ∈ V } > − ∞ .

This condition is not enough to know that a minimizer exists, but it shows the existence of a minimizing sequence, that is, a sequence ( u n ) in V such that J ( u n ) → inf { J ( u ) | u ∈ V } .

The direct method may broken into the following steps

1. Take a minimizing sequence ( u n ) for J .
2. Show that ( u n ) admits some subsequence ( u n k ) , that converges to a u 0 ∈ V with respect to a topology τ on V .
3. Show that J is sequentially lower semi-continuous with respect to the topology τ .

To see that this shows the existence of a minimizer, consider the following characterization of sequentially lower-semicontinuous functions.

The function J is sequentially lower-semicontinuous if lim inf n → ∞ J ( u n ) ≥ J ( u 0 ) for any convergent sequence u n → u 0 in V .

The conclusions follows from

inf { J ( u ) | u ∈ V } = lim n → ∞ J ( u n ) = lim k → ∞ J ( u n k ) ≥ J ( u 0 ) ≥ inf { J ( u ) | u ∈ V } ,

in other words

J ( u 0 ) = inf { J ( u ) | u ∈ V } .

Banach spaces

The direct method may often be applied with success when the space V is a subset of a separable reflexive Banach space W . In this case the sequential Banach–Alaoglu theorem implies that any bounded sequence ( u n ) in V has a subsequence that converges to some u 0 in W with respect to the weak topology. If V is sequentially closed in W , so that u 0 is in V , the direct method may be applied to a functional J : V → R ¯ by showing

1. J is bounded from below,
2. any minimizing sequence for J is bounded, and
3. J is weakly sequentially lower semi-continuous, i.e., for any weakly convergent sequence u n → u 0 it holds that lim inf n → ∞ J ( u n ) ≥ J ( u 0 ) .

The second part is usually accomplished by showing that J admits some growth condition. An example is

J ( x ) ≥ α ∥ x ∥ q − β for some α > 0 , q ≥ 1 and β ≥ 0 .

A functional with this property is sometimes called coercive. Showing sequential lower semi-continuity is usually the most difficult part when applying the direct method. See below for some theorems for a general class of functionals.

Sobolev spaces

The typical functional in the calculus of variations is an integral of the form

J ( u ) = ∫ Ω F ( x , u ( x ) , ∇ u ( x ) ) d x

where Ω is a subset of R n and F is a real-valued function on Ω × R m × R m n . The argument of J is a differentiable function u : Ω → R m , and its Jacobian ∇ u ( x ) is identified with a m n -vector.

When deriving the Euler–Lagrange equation, the common approach is to assume Ω has a C 2 boundary and let the domain of definition for J be C 2 ( Ω , R m ) . This space is a Banach space when endowed with the supremum norm, but it is not reflexive. When applying the direct method, the functional is usually defined on a Sobolev space W 1 , p ( Ω , R m ) with p > 1 , which is a reflexive Banach space. The derivatives of u in the formula for J must then be taken as weak derivatives. The next section presents two theorems regarding weak sequential lower semi-continuity of functionals of the above type.

Sequential lower semi-continuity of integrals

As many functionals in the calculus of variations are of the form

J ( u ) = ∫ Ω F ( x , u ( x ) , ∇ u ( x ) ) d x ,

where Ω ⊆ R n is open, theorems characterizing functions F for which J is weakly sequentially lower-semicontinuous in W 1 , p ( Ω , R m ) is of great importance.

In general we have the following

Assume that F is a function such that

1. The function ( y , p ) ↦ F ( x , y , p ) is continuous for almost every x ∈ Ω ,
2. the function x ↦ F ( x , y , p ) is measurable for every ( y , p ) ∈ R m × R m n , and
3. F ( x , y , p ) ≥ a ( x ) ⋅ p + b ( x ) for a fixed a ∈ L q ( Ω , R m n ) where 1 / q + 1 / p = 1 , a fixed b ∈ L 1 ( Ω ) , for a.e. x ∈ Ω and every ( y , p ) ∈ R m × R m n (here a ( x ) ⋅ p means the inner product of a ( x ) and p in R m n ).

The following holds. If the function p ↦ F ( x , y , p ) is convex for a.e. x ∈ Ω and every y ∈ R m ,then J is sequentially weakly lower semi-continuous.

When n = 1 or m = 1 the following converse-like theorem holds

Assume that F is continuous and satisfies | F ( x , y , p ) | ≤ a ( x , | y | , | p | ) for every ( x , y , p ) , and a fixed function a ( x , y , p ) increasing in y and p , and locally integrable in x . It then holds, if J is sequentially weakly lower semi-continuous, then for any given ( x , y ) ∈ Ω × R m the function p ↦ F ( x , y , p ) is convex.

In conclusion, when m = 1 or n = 1 , the functional J , assuming reasonable growth and boundedness on F , is weakly sequentially lower semi-continuous if, and only if, the function p ↦ F ( x , y , p ) is convex. If both n and m are greater than 1, it is possible to weaken the necessity of convexity to generalizations of convexity, namely polyconvexity and quasiconvexity.