Direct method in the calculus of variations

<h2 id="the-method">The method</h2>
The calculus of variations deals with functionals 
 
 
 
 J
 :
 V
 →
 
 
 
 
 R
 
 ¯
 
 
 
 
 
 {\displaystyle J:V\to {\bar {\mathbb {R} }}}
 
, where 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is some <a href="/facts/Function_space/lLUE2R1t">function space</a> and 
 
 
 
 
 
 
 
 R
 
 ¯
 
 
 
 =
 
 R
 
 ∪
 {
 ∞
 }
 
 
 {\displaystyle {\bar {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}}
 
. The main interest of the subject is to find minimizers for such functionals, that is, functions 
 
 
 
 v
 ∈
 V
 
 
 {\displaystyle v\in V}
 
 such that 
 
 
 
 J
 (
 v
 )
 ≤
 J
 (
 u
 )
 
 
 {\displaystyle J(v)\leq J(u)}
 
 for all 
 
 
 
 u
 ∈
 V
 
 
 {\displaystyle u\in V}
 
.
The standard tool for obtaining necessary conditions for a function to be a minimizer is the <a href="/facts/Euler%E2%80%93Lagrange_equation/Ymq3r3Dh">Euler–Lagrange equation</a>. 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
 
 
 {\displaystyle J}
 
 must be bounded from below to have a minimizer. This means

inf
        {
        J
        (
        u
        )
        
          |
        
        u
        ∈
        V
        }
        >
        −
        ∞
        .
        
      
    
    {\displaystyle \inf\{J(u)|u\in V\}>-\infty .\,}

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
 
 
 )
 
 
 {\displaystyle (u_{n})}
 
 in 
 
 
 
 V
 
 
 {\displaystyle V}
 
 such that 
 
 
 
 J
 (
 
 u
 
 n
 
 
 )
 →
 inf
 {
 J
 (
 u
 )
 
 |
 
 u
 ∈
 V
 }
 .
 
 
 {\displaystyle J(u_{n})\to \inf\{J(u)|u\in V\}.}

The direct method may be broken into the following steps

<ol><li>Take a minimizing sequence 
 
 
 
 (
 
 u
 
 n
 
 
 )
 
 
 {\displaystyle (u_{n})}
 
 for 
 
 
 
 J
 
 
 {\displaystyle J}
 
.</li>
<li>Show that 
 
 
 
 (
 
 u
 
 n
 
 
 )
 
 
 {\displaystyle (u_{n})}
 
 admits some <a href="/facts/Subsequence/GdC2Qwau">subsequence</a> 
 
 
 
 (
 
 u
 
 
 n
 
 k
 
 
 
 
 )
 
 
 {\displaystyle (u_{n_{k}})}
 
, that converges to a 
 
 
 
 
 u
 
 0
 
 
 ∈
 V
 
 
 {\displaystyle u_{0}\in V}
 
 with respect to a topology 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
 on 
 
 
 
 V
 
 
 {\displaystyle V}
 
.</li>
<li>Show that 
 
 
 
 J
 
 
 {\displaystyle J}
 
 is sequentially <a href="/facts/Lower_semi-continuous/4unJbWdW">lower semi-continuous</a> with respect to the topology 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
.</li></ol>
To see that this shows the existence of a minimizer, consider the following characterization of sequentially lower-semicontinuous functions.

The function 
 
 
 
 J
 
 
 {\displaystyle J}
 
 is sequentially lower-semicontinuous if

lim inf
 
 n
 →
 ∞
 
 
 J
 (
 
 u
 
 n
 
 
 )
 ≥
 J
 (
 
 u
 
 0
 
 
 )
 
 
 {\displaystyle \liminf _{n\to \infty }J(u_{n})\geq J(u_{0})}
 
 for any convergent sequence 
 
 
 
 
 u
 
 n
 
 
 →
 
 u
 
 0
 
 
 
 
 {\displaystyle u_{n}\to u_{0}}
 
 in 
 
 
 
 V
 
 
 {\displaystyle 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
 }
 
 
 {\displaystyle \inf\{J(u)|u\in V\}=\lim _{n\to \infty }J(u_{n})=\lim _{k\to \infty }J(u_{n_{k}})\geq J(u_{0})\geq \inf\{J(u)|u\in V\}}
 
,
in other words

J
 (
 
 u
 
 0
 
 
 )
 =
 inf
 {
 J
 (
 u
 )
 
 |
 
 u
 ∈
 V
 }
 
 
 {\displaystyle J(u_{0})=\inf\{J(u)|u\in V\}}
 
.
<h2 id="details">Details</h2>
<h3>Banach spaces</h3>
The direct method may often be applied with success when the space 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is a subset of a <a href="/facts/Separable_space/2bmU73bk">separable</a> <a href="/facts/Reflexive_space/g7Cjbw0S">reflexive</a> <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> 
 
 
 
 W
 
 
 {\displaystyle W}
 
. In this case the <a href="/facts/Banach%E2%80%93Alaoglu_theorem/NILk5nDm">sequential Banach–Alaoglu theorem</a> implies that any bounded sequence 
 
 
 
 (
 
 u
 
 n
 
 
 )
 
 
 {\displaystyle (u_{n})}
 
 in 
 
 
 
 V
 
 
 {\displaystyle V}
 
 has a subsequence that converges to some 
 
 
 
 
 u
 
 0
 
 
 
 
 {\displaystyle u_{0}}
 
 in 
 
 
 
 W
 
 
 {\displaystyle W}
 
 with respect to the <a href="/facts/Weak_topology/XJASE9Up">weak topology</a>. If 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is sequentially closed in 
 
 
 
 W
 
 
 {\displaystyle W}
 
, so that 
 
 
 
 
 u
 
 0
 
 
 
 
 {\displaystyle u_{0}}
 
 is in 
 
 
 
 V
 
 
 {\displaystyle V}
 
, the direct method may be applied to a functional 
 
 
 
 J
 :
 V
 →
 
 
 
 
 R
 
 ¯
 
 
 
 
 
 {\displaystyle J:V\to {\bar {\mathbb {R} }}}
 
 by showing

<ol><li>
 
 
 
 J
 
 
 {\displaystyle J}
 
 is bounded from below,</li>
<li>any minimizing sequence for 
 
 
 
 J
 
 
 {\displaystyle J}
 
 is bounded, and</li>
<li>
 
 
 
 J
 
 
 {\displaystyle J}
 
 is weakly sequentially lower semi-continuous, i.e., for any weakly convergent sequence 
 
 
 
 
 u
 
 n
 
 
 →
 
 u
 
 0
 
 
 
 
 {\displaystyle u_{n}\to u_{0}}
 
 it holds that 
 
 
 
 
 lim inf
 
 n
 →
 ∞
 
 
 J
 (
 
 u
 
 n
 
 
 )
 ≥
 J
 (
 
 u
 
 0
 
 
 )
 
 
 {\displaystyle \liminf _{n\to \infty }J(u_{n})\geq J(u_{0})}
 
.</li></ol>
The second part is usually accomplished by showing that 
 
 
 
 J
 
 
 {\displaystyle J}
 
 admits some growth condition. An example is

J
 (
 x
 )
 ≥
 α
 ‖
 x
 
 ‖
 
 q
 
 
 −
 β
 
 
 {\displaystyle J(x)\geq \alpha \lVert x\rVert ^{q}-\beta }
 
 for some 
 
 
 
 α
 >
 0
 
 
 {\displaystyle \alpha >0}
 
, 
 
 
 
 q
 ≥
 1
 
 
 {\displaystyle q\geq 1}
 
 and 
 
 
 
 β
 ≥
 0
 
 
 {\displaystyle \beta \geq 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.

<h3>Sobolev spaces</h3>
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
      
    
    {\displaystyle J(u)=\int _{\Omega }F(x,u(x),\nabla u(x))dx}

where 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 is a subset of 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 and 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is a real-valued function on 
 
 
 
 Ω
 ×
 
 
 R
 
 
 m
 
 
 ×
 
 
 R
 
 
 m
 n
 
 
 
 
 {\displaystyle \Omega \times \mathbb {R} ^{m}\times \mathbb {R} ^{mn}}
 
. The argument of 
 
 
 
 J
 
 
 {\displaystyle J}
 
 is a differentiable function 
 
 
 
 u
 :
 Ω
 →
 
 
 R
 
 
 m
 
 
 
 
 {\displaystyle u:\Omega \to \mathbb {R} ^{m}}
 
, and its <a href="/facts/Jacobian_matrix_and_determinant/0tNv6o8j">Jacobian</a> 
 
 
 
 ∇
 u
 (
 x
 )
 
 
 {\displaystyle \nabla u(x)}
 
 is identified with a 
 
 
 
 m
 n
 
 
 {\displaystyle mn}
 
-vector.
When deriving the Euler–Lagrange equation, the common approach is to assume 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 has a 
 
 
 
 
 C
 
 2
 
 
 
 
 {\displaystyle C^{2}}
 
 boundary and let the domain of definition for 
 
 
 
 J
 
 
 {\displaystyle J}
 
 be 
 
 
 
 
 C
 
 2
 
 
 (
 Ω
 ,
 
 
 R
 
 
 m
 
 
 )
 
 
 {\displaystyle C^{2}(\Omega ,\mathbb {R} ^{m})}
 
. This space is a Banach space when endowed with the <a href="/facts/Supremum_norm/5yIkjNE3">supremum norm</a>, but it is not reflexive. When applying the direct method, the functional is usually defined on a <a href="/facts/Sobolev_space/34qxWCyr">Sobolev space</a> 
 
 
 
 
 W
 
 1
 ,
 p
 
 
 (
 Ω
 ,
 
 
 R
 
 
 m
 
 
 )
 
 
 {\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})}
 
 with 
 
 
 
 p
 >
 1
 
 
 {\displaystyle p>1}
 
, which is a reflexive Banach space. The derivatives of 
 
 
 
 u
 
 
 {\displaystyle u}
 
 in the formula for 
 
 
 
 J
 
 
 {\displaystyle J}
 
 must then be taken as <a href="/facts/Weak_derivative/oehSmSMg">weak derivatives</a>.
Another common function space is 
 
 
 
 
 W
 
 g
 
 
 1
 ,
 p
 
 
 (
 Ω
 ,
 
 
 R
 
 
 m
 
 
 )
 
 
 {\displaystyle W_{g}^{1,p}(\Omega ,\mathbb {R} ^{m})}
 
 which is the affine sub space of 
 
 
 
 
 W
 
 1
 ,
 p
 
 
 (
 Ω
 ,
 
 
 R
 
 
 m
 
 
 )
 
 
 {\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})}
 
 of functions whose <a href="/facts/Trace_operator/7uQ4kSE4">trace</a> is some fixed function 
 
 
 
 g
 
 
 {\displaystyle g}
 
 in the image of the trace operator. This restriction allows finding minimizers of the functional 
 
 
 
 J
 
 
 {\displaystyle J}
 
 that satisfy some desired boundary conditions. This is similar to solving the Euler–Lagrange equation with Dirichlet boundary conditions. Additionally there are settings in which there are minimizers in 
 
 
 
 
 W
 
 g
 
 
 1
 ,
 p
 
 
 (
 Ω
 ,
 
 
 R
 
 
 m
 
 
 )
 
 
 {\displaystyle W_{g}^{1,p}(\Omega ,\mathbb {R} ^{m})}
 
 but not in 
 
 
 
 
 W
 
 1
 ,
 p
 
 
 (
 Ω
 ,
 
 
 R
 
 
 m
 
 
 )
 
 
 {\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})}
 
.
The idea of solving minimization problems while restricting the values on the boundary can be further generalized by looking on function spaces where the trace is fixed only on a part of the boundary, and can be arbitrary on the rest.
The next section presents theorems regarding weak sequential lower semi-continuity of functionals of the above type.

<h2 id="sequential-lower-semi-continuity-of-integrals">Sequential lower semi-continuity of integrals</h2>
As many functionals in the calculus of variations are of the form

J
 (
 u
 )
 =
 
 ∫
 
 Ω
 
 
 F
 (
 x
 ,
 u
 (
 x
 )
 ,
 ∇
 u
 (
 x
 )
 )
 d
 x
 
 
 {\displaystyle J(u)=\int _{\Omega }F(x,u(x),\nabla u(x))dx}
 
,
where 
 
 
 
 Ω
 ⊆
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \Omega \subseteq \mathbb {R} ^{n}}
 
 is open, theorems characterizing functions 
 
 
 
 F
 
 
 {\displaystyle F}
 
 for which 
 
 
 
 J
 
 
 {\displaystyle J}
 
 is weakly sequentially lower-semicontinuous in 
 
 
 
 
 W
 
 1
 ,
 p
 
 
 (
 Ω
 ,
 
 
 R
 
 
 m
 
 
 )
 
 
 {\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})}
 
 with 
 
 
 
 p
 ≥
 1
 
 
 {\displaystyle p\geq 1}
 
 is of great importance.
In general one has the following:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

Assume that 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is a function that has the following properties:
<ol><li>The function 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is a <a href="/facts/Carath%C3%A9odory_function/mUdHxT4C">Carathéodory function</a>.</li>
<li>There exist 
 
 
 
 a
 ∈
 
 L
 
 q
 
 
 (
 Ω
 ,
 
 
 R
 
 
 m
 n
 
 
 )
 
 
 {\displaystyle a\in L^{q}(\Omega ,\mathbb {R} ^{mn})}
 
 with <a href="/facts/H%C3%B6lder_conjugate/7I1ubTWw">Hölder conjugate</a> 
 
 
 
 q
 =
 
 
 
 p
 
 p
 −
 1
 
 
 
 
 
 
 {\displaystyle q={\tfrac {p}{p-1}}}
 
 and 
 
 
 
 b
 ∈
 
 L
 
 1
 
 
 (
 Ω
 )
 
 
 {\displaystyle b\in L^{1}(\Omega )}
 
 such that the following inequality holds true for almost every 
 
 
 
 x
 ∈
 Ω
 
 
 {\displaystyle x\in \Omega }
 
 and every 
 
 
 
 (
 y
 ,
 A
 )
 ∈
 
 
 R
 
 
 m
 
 
 ×
 
 
 R
 
 
 m
 n
 
 
 
 
 {\displaystyle (y,A)\in \mathbb {R} ^{m}\times \mathbb {R} ^{mn}}
 
: 
 
 
 
 F
 (
 x
 ,
 y
 ,
 A
 )
 ≥
 ⟨
 a
 (
 x
 )
 ,
 A
 ⟩
 +
 b
 (
 x
 )
 
 
 {\displaystyle F(x,y,A)\geq \langle a(x),A\rangle +b(x)}
 
. Here, 
 
 
 
 ⟨
 a
 (
 x
 )
 ,
 A
 ⟩
 
 
 {\displaystyle \langle a(x),A\rangle }
 
 denotes the <a href="/facts/Frobenius_inner_product/E8DB5N9E">Frobenius inner product</a> of 
 
 
 
 a
 (
 x
 )
 
 
 {\displaystyle a(x)}
 
 and 
 
 
 
 A
 
 
 {\displaystyle A}
 
 in 
 
 
 
 
 
 R
 
 
 m
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{mn}}
 
).</li></ol>
If the function 
 
 
 
 A
 ↦
 F
 (
 x
 ,
 y
 ,
 A
 )
 
 
 {\displaystyle A\mapsto F(x,y,A)}
 
 is convex for almost every 
 
 
 
 x
 ∈
 Ω
 
 
 {\displaystyle x\in \Omega }
 
 and every 
 
 
 
 y
 ∈
 
 
 R
 
 
 m
 
 
 
 
 {\displaystyle y\in \mathbb {R} ^{m}}
 
,
then 
 
 
 
 J
 
 
 {\displaystyle J}
 
 is sequentially weakly lower semi-continuous.
When 
 
 
 
 n
 =
 1
 
 
 {\displaystyle n=1}
 
 or 
 
 
 
 m
 =
 1
 
 
 {\displaystyle m=1}
 
 the following converse-like theorem holds<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

Assume that 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is continuous and satisfies

|
        
        F
        (
        x
        ,
        y
        ,
        A
        )
        
          |
        
        ≤
        a
        (
        x
        ,
        
          |
        
        y
        
          |
        
        ,
        
          |
        
        A
        
          |
        
        )
      
    
    {\displaystyle |F(x,y,A)|\leq a(x,|y|,|A|)}

for every 
 
 
 
 (
 x
 ,
 y
 ,
 A
 )
 
 
 {\displaystyle (x,y,A)}
 
, and a fixed function 
 
 
 
 a
 (
 x
 ,
 
 |
 
 y
 
 |
 
 ,
 
 |
 
 A
 
 |
 
 )
 
 
 {\displaystyle a(x,|y|,|A|)}
 
 increasing in 
 
 
 
 
 |
 
 y
 
 |
 
 
 
 {\displaystyle |y|}
 
 and 
 
 
 
 
 |
 
 A
 
 |
 
 
 
 {\displaystyle |A|}
 
, and locally integrable in 
 
 
 
 x
 
 
 {\displaystyle x}
 
. If 
 
 
 
 J
 
 
 {\displaystyle J}
 
 is sequentially weakly lower semi-continuous, then for any given 
 
 
 
 (
 x
 ,
 y
 )
 ∈
 Ω
 ×
 
 
 R
 
 
 m
 
 
 
 
 {\displaystyle (x,y)\in \Omega \times \mathbb {R} ^{m}}
 
 the function 
 
 
 
 A
 ↦
 F
 (
 x
 ,
 y
 ,
 A
 )
 
 
 {\displaystyle A\mapsto F(x,y,A)}
 
 is convex.
In conclusion, when 
 
 
 
 m
 =
 1
 
 
 {\displaystyle m=1}
 
 or 
 
 
 
 n
 =
 1
 
 
 {\displaystyle n=1}
 
, the functional 
 
 
 
 J
 
 
 {\displaystyle J}
 
, assuming reasonable growth and boundedness on 
 
 
 
 F
 
 
 {\displaystyle F}
 
, is weakly sequentially lower semi-continuous if, and only if the function 
 
 
 
 A
 ↦
 F
 (
 x
 ,
 y
 ,
 A
 )
 
 
 {\displaystyle A\mapsto F(x,y,A)}
 
 is convex.
However, there are many interesting cases where one cannot assume that 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is convex. The following theorem<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> proves sequential lower semi-continuity using a weaker notion of convexity:

Assume that 
 
 
 
 F
 :
 Ω
 ×
 
 
 R
 
 
 m
 
 
 ×
 
 
 R
 
 
 m
 n
 
 
 →
 [
 0
 ,
 ∞
 )
 
 
 {\displaystyle F:\Omega \times \mathbb {R} ^{m}\times \mathbb {R} ^{mn}\to [0,\infty )}
 
 is a function that has the following properties:
<ol><li>The function 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is a <a href="/facts/Carath%C3%A9odory_function/mUdHxT4C">Carathéodory function</a>.</li>
<li>The function 
 
 
 
 F
 
 
 {\displaystyle F}
 
 has 
 
 
 
 p
 
 
 {\displaystyle p}
 
-growth for some 
 
 
 
 p
 >
 1
 
 
 {\displaystyle p>1}
 
: There exists a constant 
 
 
 
 C
 
 
 {\displaystyle C}
 
 such that for every 
 
 
 
 y
 ∈
 
 
 R
 
 
 m
 
 
 
 
 {\displaystyle y\in \mathbb {R} ^{m}}
 
 and for <a href="/facts/Almost_every/1quRFtJE">almost every</a> 
 
 
 
 x
 ∈
 Ω
 
 
 {\displaystyle x\in \Omega }
 
 
 
 
 
 
 |
 
 F
 (
 x
 ,
 y
 ,
 A
 )
 
 |
 
 ≤
 C
 (
 1
 +
 
 |
 
 y
 
 
 |
 
 
 p
 
 
 +
 
 |
 
 A
 
 
 |
 
 
 p
 
 
 )
 
 
 {\displaystyle |F(x,y,A)|\leq C(1+|y|^{p}+|A|^{p})}
 
.</li>
<li>For every 
 
 
 
 y
 ∈
 
 
 R
 
 
 m
 
 
 
 
 {\displaystyle y\in \mathbb {R} ^{m}}
 
 and for <a href="/facts/Almost_every/1quRFtJE">almost every</a> 
 
 
 
 x
 ∈
 Ω
 
 
 {\displaystyle x\in \Omega }
 
, the function 
 
 
 
 A
 ↦
 F
 (
 x
 ,
 y
 ,
 A
 )
 
 
 {\displaystyle A\mapsto F(x,y,A)}
 
 is quasiconvex: there exists a cube 
 
 
 
 D
 ⊆
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle D\subseteq \mathbb {R} ^{n}}
 
 such that for every 
 
 
 
 A
 ∈
 
 
 R
 
 
 m
 n
 
 
 ,
 φ
 ∈
 
 W
 
 0
 
 
 1
 ,
 ∞
 
 
 (
 Ω
 ,
 
 
 R
 
 
 m
 
 
 )
 
 
 {\displaystyle A\in \mathbb {R} ^{mn},\varphi \in W_{0}^{1,\infty }(\Omega ,\mathbb {R} ^{m})}
 
 it holds:</li></ol>

 
 
 
 F
 (
 x
 ,
 y
 ,
 A
 )
 ≤
 
 |
 
 D
 
 
 |
 
 
 −
 1
 
 
 
 ∫
 
 D
 
 
 F
 (
 x
 ,
 y
 ,
 A
 +
 ∇
 φ
 (
 z
 )
 )
 d
 z
 
 
 {\displaystyle F(x,y,A)\leq |D|^{-1}\int _{D}F(x,y,A+\nabla \varphi (z))dz}

where 
 
 
 
 
 |
 
 D
 
 |
 
 
 
 {\displaystyle |D|}
 
 is the <a href="/facts/Volume/aeAXCBUd">volume</a> of 
 
 
 
 D
 
 
 {\displaystyle D}
 
.
Then 
 
 
 
 J
 
 
 {\displaystyle J}
 
 is sequentially weakly lower semi-continuous in 
 
 
 
 
 W
 
 1
 ,
 p
 
 
 (
 Ω
 ,
 
 
 R
 
 
 m
 
 
 )
 
 
 {\displaystyle W^{1,p}(\Omega ,\mathbb {R} ^{m})}
 
.
A converse like theorem in this case is the following:
<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

Assume that 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is continuous and satisfies

<h2 id="references-and-further-reading">References and further reading</h2>
<ul><li>Dacorogna, Bernard (1989). Direct Methods in the Calculus of Variations. Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-50491-5.</li>
<li><a href="/facts/Irene_Fonseca/Ps7up8B7">Fonseca, Irene</a>; Giovanni Leoni (2007). Modern Methods in the Calculus of Variations: 
 
 
 
 
 L
 
 p
 
 
 
 
 {\displaystyle L^{p}}
 
 Spaces. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-35784-3.</li>
<li>Morrey, C. B., Jr.: Multiple Integrals in the Calculus of Variations. Springer, 1966 (reprinted 2008), Berlin <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-69915-6.</li>
<li>Jindřich Nečas: Direct Methods in the Theory of Elliptic Equations. (Transl. from French original 1967 by A.Kufner and G.Tronel), Springer, 2012, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-10455-8.</li>
<li>T. Roubíček (2000). "Direct method for parabolic problems". Adv. Math. Sci. Appl. Vol. 10. pp. 57–65. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1769181">1769181</a>.</li>
<li>Acerbi Emilio, Fusco Nicola. "Semicontinuity problems in the calculus of variations." Archive for Rational Mechanics and Analysis 86.2 (1984): 125-145</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Dacorogna, pp. 1–43. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">I. M. Gelfand; S. V. Fomin (1991). Calculus of Variations. Dover Publications. ISBN 978-0-486-41448-5. <a href="978-0-486-41448-5" target="_blank">978-0-486-41448-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Dacorogna, pp. 74–79. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Dacorogna, pp. 66–74. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Acerbi-Fusco <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Dacorogna, pp. 156. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Direct method in the calculus of variations open-in-new

Direct method in the calculus of variations