Direct method in the calculus of variations

The direct method takes a minimizing sequence, extracts a subsequence using compactness, and uses lower semicontinuity to show that its limit attains the infimum.

Suppose the domain has finite measure and $F\in C^1$ is convex, with $|F(p)|\le C(1+|p{|}^2)$ and $|DF(p)|\le C(1+|p|)$. The supporting inequality $F(p)\ge F(q)+DF(q)\cdot (p-q)$, integrated with $q=Du$, gives $\int F(D{u}_k)\ge \int F(Du)+\int DF(Du)\cdot (D{u}_k-Du)$. The last term tends to zero by weak convergence of the gradients and the fixed $L^2$ test field $DF(Du)$. This proves weak lower semicontinuity without requiring pointwise convergence of gradients.

For a bounded domain, bounded symmetric uniformly elliptic $A$, $q\le 0$, and $f\in L^2$, minimizing $J$ on $\psi +H_0^1$ solves $\mathrm{div}(A\nabla u)+qu=f$. The nonnegative quadratic terms are weakly lower semicontinuous, and Poincaré inequality makes the gradient norm coercive on the zero-boundary part. The first variation gives the weak equation with the forcing sign displayed above. If $q$ is positive, coercivity may fail at a Dirichlet eigenvalue; orthogonality to the homogeneous kernel is then needed for solvability.

For real $f\in L^2({\mathbb{R}}^3)$, this energy functional on $H^1({\mathbb{R}}^3)$ is coercive and attains its minimum by the direct method in the calculus of variations. The nonnegative potential term is weakly lower semicontinuous by local Sobolev compactness gives lower semicontinuity of a nonnegative integral. Its Euler-Lagrange equation is $-\Delta u+u+\sin u=f$ in the weak solution sense. The extra linear restoring term screens the Sine-Gordon equation nonlinearity. The scalar potential $V(s)=s^2/2+1-\cos s$ is convex because $V^{\prime \prime }(s)=1+\cos s\ge 0$; hence any weak critical point is a minimizer. The elliptic regularity estimate for $-\Delta +1$, together with $|\sin u|\le |u|$, gives $u\in H^2$. The Sobolev inequality then puts $u$ in $W^{1,6}$, so Morrey's inequality and uniformly continuous integrable functions vanish at infinity give a continuous representative tending to zero.

Take $a=3\pi /2$, $L=10$ and $u(x)=a\exp (-|x{|}^2/L^2)$ on ${\mathbb{R}}^3$. Set $f=-\Delta u+u+\sin u$. Both functions are smooth and belong to L2 space, and $u\in H^1$. The strictly increasing reaction $h(s)=s+\sin s$ has $0\le h(u)\le h(a)=a-1$, whileConsequently $\Vert f{\Vert }_{\infty }\le a-1+6a/L^2<a=\Vert u{\Vert }_{\infty }$. This $u$ is a minimizer of the screened sine-Gordon energy: it solves the Euler-Lagrange equation and that energy is convex. Thus even the minimizing and smooth hypotheses do not justify the stronger source-size bound. The appropriate general estimate is maximum bound for a monotone reaction term.

A minimizing sequence satisfies $E(u_n)\to \inf E$. Coercivity bounds an appropriate sublevel set, Compactness supplies a convergent subsequence, and sequential lower semicontinuity turns its limit into a global minimizer.

Boundary data $g$ satisfy the bounded slope condition with constant $K$ when every boundary point admits affine lower and upper supporting functions of Lipschitz constant at most $K$. Comparison with these barriers gives a global Lipschitz bound for convex variational problems.

For suitable convex integral functionals, ordered boundary values of two minimizers imply the same ordering in the domain. Affine minimizers can therefore serve as upper and lower barriers.