The Direct Method and Euler--Lagrange Equations — Examples

Given: The functional $I:{\mathbb{R}}^2\to \mathbb{R}$, $I(x,y)=x^2$, on the real Banach space ${\mathbb{R}}^2$.

A function $I$ on a convex set is convex when $I(\lambda u+(1-\lambda )v)\le \lambda I(u)+(1-\lambda )I(v)$ for all $u,v$ and $\lambda \in [0,1]$, and strictly convex when the inequality is strict for $u\mathrm{\ne }v$ and $\lambda \in (0,1)$ with finite values (Convex and strictly convex functionals on a convex subset of a real vector space).

A proper, strictly convex functional has at most one minimiser on a convex set; in particular at most one finite minimiser (Strict convexity gives uniqueness of a minimiser).

Given: The Axiom of Choice; the real Hilbert space $H={\mathrm{\ell }}^2(\mathbb{N},\mathbb{R})$ (Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$), its coordinate vectors $e_k$, namely the family that is $1$ at $k$ and $0$ elsewhere, and the functional $I:H\to \mathbb{R}$ with $I(0):=1$ and $I(u):=\mathrm{\parallel }u{\mathrm{\parallel }}^2$ for $u\mathrm{\ne }0$. The counting-measure dictionary ${\mathrm{\ell }}^p$ is the $L^p$ space of counting measure identifies $H$ with real $L^2(\mathrm{\#})$, which is reflexive (and thus Banach) by Reflexivity of Lp for one less p less infinity under Countable Choice, supplied here by AC; the series pairing is its inner product.

Proper, coercive and weakly sequentially lower semicontinuous functionals are defined as in Proper, coercive and weakly lower semicontinuous extended-real functionals; a functional is proper when its effective domain is nonempty, equivalently when its infimum is less than $+\mathrm{\infty }$ (it may be $-\mathrm{\infty }$).

In the direct method, weak sequential lower semicontinuity is a hypothesis alongside coercivity; The direct method in a reflexive Banach space states all of its hypotheses explicitly, so a coercive proper functional on a reflexive space need not attain when that hypothesis fails.

Each coordinate vector satisfies $e_k\in H$ and $\mathrm{\parallel }{e}_k\mathrm{\parallel }=1$, and $e_k\rightharpoonup 0$: by the duality of ${\mathrm{\ell }}^p$ and ${\mathrm{\ell }}^q$ every bounded linear functional on ${\mathrm{\ell }}^2$ is $\mathrm{\Lambda }(a)={\sum }_k{a}_k{b}_k$ for a unique $b\in {\mathrm{\ell }}^2$ (Counting measure specializes the representation theorem to ${\mathrm{\ell }}^p$ and ${\mathrm{\ell }}^q$), so $\mathrm{\Lambda }(e_k)=b_k$, and $b_k\to 0$ because a square-summable family has small tails, that is, for every $\epsilon >0$ there is a finite $F$ with ${\sum }_{k\mathrm{\notin }F}\mathrm{\mid }{b}_k{\mathrm{\mid }}^2<\epsilon$ (Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$), whence $\mathrm{\mid }{b}_k{\mathrm{\mid }}^2<\epsilon$ for every $k$ beyond all elements of $F$; weak convergence means convergence against every bounded linear functional (Weak convergence of nets and sequences).

Given: The Axiom of Choice; the real Hilbert space $H={\mathrm{\ell }}^2(\mathbb{N},\mathbb{R})$ (Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$), its coordinate vectors $e_k$, namely the family that is $1$ at $k$ and $0$ elsewhere, its closed unit ball $B=\{u\in H:\mathrm{\parallel }u\mathrm{\parallel }\le 1\}$, and the functional $I(u)={\sum }_{j\ge 0}{u}_j^2\mathrm{/}(j+1)$. The counting-measure dictionary ${\mathrm{\ell }}^p$ is the $L^p$ space of counting measure identifies $H$ with real $L^2(\mathrm{\#})$, which is reflexive (and thus Banach) by Reflexivity of Lp for one less p less infinity under Countable Choice, supplied here by AC; the series pairing is its inner product.

The series defining $I$ converges absolutely for every $u\in H$ because $0\le u_j^2\mathrm{/}(j+1)\le u_j^2$ and ${\sum }_j{u}_j^2=\mathrm{\parallel }u{\mathrm{\parallel }}^2<\mathrm{\infty }$; moreover $\mathrm{\mid }I(u+v)-I(u)\mathrm{\mid }\le 2\mathrm{\parallel }u\mathrm{\parallel }\mathrm{\parallel }v\mathrm{\parallel }+\mathrm{\parallel }v{\mathrm{\parallel }}^2$, so $I$ is continuous, and $I$ is convex and nonnegative (Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$).

Each coordinate vector satisfies $e_k\in H$, $\mathrm{\parallel }{e}_k\mathrm{\parallel }=1$ and $e_k\rightharpoonup 0$: by the duality of ${\mathrm{\ell }}^p$ and ${\mathrm{\ell }}^q$ every bounded linear functional on ${\mathrm{\ell }}^2$ is $\mathrm{\Lambda }(a)={\sum }_k{a}_k{b}_k$ for a unique $b\in {\mathrm{\ell }}^2$ (Counting measure specializes the representation theorem to ${\mathrm{\ell }}^p$ and ${\mathrm{\ell }}^q$), so $\mathrm{\Lambda }(e_k)=b_k$, and $b_k\to 0$ because a square-summable family has small tails, that is, for every $\epsilon >0$ there is a finite $F$ with ${\sum }_{k\mathrm{\notin }F}\mathrm{\mid }{b}_k{\mathrm{\mid }}^2<\epsilon$ (Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$), whence $\mathrm{\mid }{b}_k{\mathrm{\mid }}^2<\epsilon$ for every $k$ beyond all elements of $F$; weak convergence means convergence against every bounded linear functional (Weak convergence of nets and sequences).

In the direct method the compactness recovered is weak sequential compactness: a norm-bounded sequence in a reflexive Banach space has a weakly convergent subsequence (A bounded sequence in a reflexive Banach space has a weakly convergent subsequence), and the limit need not be a strong limit (The direct method in a reflexive Banach space, Proper, coercive and weakly lower semicontinuous extended-real functionals).

Given: The Axiom of Choice; the real Hilbert space $H={\mathrm{\ell }}^2(\mathbb{N},\mathbb{R})$ (Hilbert space, Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$), its coordinate vectors $e_j$, the families that are $1$ at $j$ and $0$ elsewhere, and its unit sphere $S=\{u\in H:\mathrm{\parallel }u\mathrm{\parallel }=1\}$. The counting-measure dictionary ${\mathrm{\ell }}^p$ is the $L^p$ space of counting measure identifies $H$ with real $L^2(\mathrm{\#})$, which is reflexive (and thus Banach) by Reflexivity of Lp for one less p less infinity under Countable Choice, supplied here by AC; the series pairing is its inner product.

The unit sphere of an infinite-dimensional Hilbert space is norm closed and norm bounded; its weak closure is the closed unit ball (Weak closure of the unit sphere is the closed unit ball).

Each coordinate vector satisfies $e_j\in H$, $\mathrm{\parallel }{e}_j\mathrm{\parallel }=1$ and $e_j\rightharpoonup 0$: by the duality of ${\mathrm{\ell }}^p$ and ${\mathrm{\ell }}^q$ every bounded linear functional on ${\mathrm{\ell }}^2$ is $\mathrm{\Lambda }(a)={\sum }_j{a}_j{b}_j$ for a unique $b\in {\mathrm{\ell }}^2$ (Counting measure specializes the representation theorem to ${\mathrm{\ell }}^p$ and ${\mathrm{\ell }}^q$), so $\mathrm{\Lambda }(e_j)=b_j$, and $b_j\to 0$ because a square-summable family has small tails, that is, for every $\epsilon >0$ there is a finite $F$ with ${\sum }_{j\mathrm{\notin }F}\mathrm{\mid }{b}_j{\mathrm{\mid }}^2<\epsilon$ (Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$), whence $\mathrm{\mid }{b}_j{\mathrm{\mid }}^2<\epsilon$ for every $j$ beyond all elements of $F$; weak convergence means convergence against every bounded linear functional (Weak convergence of nets and sequences). Hence the sequence $(e_j)$ lies in $S$ and converges weakly to the origin, which is not in $S$.

A norm-closed convex set is weakly closed under the Axiom of Choice (A norm-closed convex set is weakly sequentially closed); the direct method's admissibility requirement is weak sequential closedness (The direct method in a reflexive Banach space).

Given: The Axiom of Choice; a bounded $C^1$ domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, the functional $J(u)=-\frac{1}{2}{\int }_{\mathrm{\Omega }}\mathrm{\mid }Du{\mathrm{\mid }}^{2}dx$ on $H_0^1(\mathrm{\Omega })$, and the Lagrangian $f(x,s,\xi )=-\frac{1}{2}\mathrm{\mid }\xi {\mathrm{\mid }}^2$.

A stationary point of $J$ is a point $u\in H_0^1(\mathrm{\Omega })$ whose first variation vanishes in every direction $\phi \in H_0^1(\mathrm{\Omega })$. The quadratic expansion below computes this variation directly for every $n\ge 1$. For $n\ge 2$, its vanishing is also the fixed-zero-trace weak Euler-Lagrange formula of The weak Euler-Lagrange equation for integral functionals with fixed trace, since $f_s=0$, $f_{\xi }=-\xi$ and the $p=2$ differentiation growth bounds hold.

On the full linear space $H_0^1(\mathrm{\Omega })$, at a local minimiser the first variation vanishes (The first variation vanishes at an interior minimiser); the converse requires convexity and stationarity in the sense $\delta I(u;v-u)\ge 0$ for all competitors, by Stationarity is sufficient for a global minimum of a convex differentiable functional.

Poincare's inequality controls the $L^2$ norm by the Dirichlet energy on $H_0^1(\mathrm{\Omega })$ (The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction); in particular ${\int }_{\mathrm{\Omega }}\mathrm{\mid }Du{\mathrm{\mid }}^2=0$ forces $u=0$ almost everywhere.

The Lagrangian $f(x,s,\xi )=-\frac{1}{2}\mathrm{\mid }\xi {\mathrm{\mid }}^2$ is concave, not convex, in $\xi$ (Convex and strictly convex functionals on a convex subset of a real vector space).

Given: Countable Choice; a function $V\in C^1(\mathbb{R})$, a compact interval $[a,b]\subset \mathbb{R}$ with $a<b$, the functional $I(u)={\int }_a^b(\frac{1}{2}{u}^{\mathrm{\prime }}(x{)}^2+V(u(x)))dx$ on the admissible class of $u\in C^2([a,b])$ with fixed endpoint values $u(a)=u_0$, $u(b)=u_1$, and the Lagrangian $f(x,s,\xi )=\frac{1}{2}{\xi }^2+V(s)$.

The conclusion has the shape of the classical Euler-Lagrange equation of The classical Euler-Lagrange equation under regularity for the Lagrangian $f(x,s,\xi )=\frac{1}{2}{\xi }^2+V(s)$, whose partials are $f_{\xi }=\xi$ and $f_s=V^{\mathrm{\prime }}(s)$. That corollary also assumes $f\in C^2$ and the growth hypotheses of the weak Euler-Lagrange theorem (The weak Euler-Lagrange equation for integral functionals with fixed trace), which need not hold for a general $V\in C^1$; the verification below therefore computes the equation directly from the minimality of $u$.

One-variable mean value theorem and Fermat's interior-extremum theorem: a differentiable function on an interval with an interior local extremum has vanishing derivative there, and the mean value theorem identifies $(V(u+\epsilon \phi )-V(u))\mathrm{/}\epsilon$ with $V^{\mathrm{\prime }}(u+\theta \epsilon \phi )\phi$ for some $\theta \in (0,1)$ (The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$, Fermat's interior extremum theorem: if $f$ has a local extremum at a point $c$ interior to its domain and is differentiable at $c$, then $f^{\mathrm{\prime }}(c)=0$).

Integration by parts on a compactly supported test function: ${\int }_a^b{u}^{\mathrm{\prime }}{\phi }^{\mathrm{\prime }}dx=-{\int }_a^b{u}^{\mathrm{\prime }\mathrm{\prime }}\phi dx$ for $\phi \in C_c^{\mathrm{\infty }}((a,b))$ and $u\in C^2([a,b])$. This is Integration by parts for continuous factors with Riemann-integrable extensions of their interior derivatives with $F=u^{\mathrm{\prime }}$ and $G=\phi$; its continuous integrands have equal Riemann and Lebesgue integrals by A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral.

Fundamental lemma: a continuous function on an open set orthogonal to every compactly supported smooth test function vanishes identically (The fundamental lemma of the calculus of variations).

A $C^2$ function on $[a,b]$ with $u^{\mathrm{\prime }\mathrm{\prime }}=0$ on $(a,b)$ is affine, since $u^{\mathrm{\prime }}$ has vanishing derivative and is therefore constant (The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$).

Given: The Axiom of Choice (for ACL), the ultrafilter lemma, DC and HB; the interval $\mathrm{\Omega }=(0,1)$; the $1$-periodic continuous function $w$ with $w(t)=t$ on $[0,\frac{1}{2}]$ and $w(t)=1-t$ on $[\frac{1}{2},1]$; the functions $u_k(x)=\frac{1}{k}w(kx)$; and the integrand $f(\xi )=({\xi }^2-1{)}^2$ with $I(u)={\int }_0^1(u^{\mathrm{\prime }}(x{)}^2-1{)}^{2}dx$. The weak compactness step uses the ultrafilter lemma, DC and HB (The ultrafilter extension principle (UL/BPI), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain, The real dominated-extension principle as an additional hypothesis over ZF).

The function $w$ is continuous, $1$-periodic, satisfies $\mathrm{\mid }w\mathrm{\mid }\le \frac{1}{2}$ and $w(0)=w(1)=0$, and is piecewise linear with $\mathrm{\mid }{w}^{\mathrm{\prime }}\mathrm{\mid }=1$ almost everywhere. Hence $u_k(x)=\frac{1}{k}w(kx)$ is absolutely continuous on $[0,1]$ with a.e. derivative $u_k^{\mathrm{\prime }}(x)=w^{\mathrm{\prime }}(kx)\in \{\pm 1\}$; by the ACL characterisation $u_k\in W^{1,2}(0,1)$ with weak derivative $u_k^{\mathrm{\prime }}$ (The ACL characterisation of $W^{1,p}$, Integer-order Sobolev spaces and their norms).

For $\phi \in L^2(0,1)$ the maps $u\mapsto {\int }_0^{1}u\phi$ and $u\mapsto {\int }_0^1{u}^{\mathrm{\prime }}\phi$ are bounded linear functionals on $W^{1,2}(0,1)$ with operator norm at most $\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^2}$, by Cauchy-Schwarz against the two components of the Sobolev norm (Integer-order Sobolev spaces and their norms).

$W^{1,2}(0,1)$ is a reflexive Banach space, so every norm-bounded sequence in it has a weakly convergent subsequence under the ultrafilter lemma, DC and HB (W^{1,p}(Omega) is reflexive for 1<p<infinity, Reflexivity is equivalent to weak subsequential compactness of bounded sequences).

If $y\in L^2(0,1;\mathbb{R})$ has $\int y\phi =0$ for all real $\phi \in L^2$, take $\phi =y$ to obtain $\mathrm{\parallel }y{\mathrm{\parallel }}_2^2=0$, so $y=0$ as an $L^2$ class. This directly identifies the subsequential Sobolev limit; uniqueness of weak probability-measure limits is not used.

The weak lower semicontinuity lemma assumes convexity of the functional (A convex norm-lower-semicontinuous functional is weakly lower semicontinuous). The convex integral existence theorem also assumes $n\ge 2$ and a $p$-growth upper bound (The direct method for convex integral functionals); the present interval and quartic integrand fail these hypotheses for $p=2$. Nonconvexity is checked by the midpoint inequality (Convex and strictly convex functionals on a convex subset of a real vector space), and existence here is decided by the explicit values of $I$.