  We present a method for finding lower bounds on the global infima of integral
variational problems, wherein $\int_\Omega f(x,u(x),\nabla u(x)){\rm d}x$ is
minimized over functions $u\colon\Omega\subset\mathbb{R}^n\to\mathbb{R}^m$
satisfying given equality or inequality constraints. Each constraint may be
imposed over $\Omega$ or its boundary, either pointwise or in an integral
sense. These global minimizations are generally non-convex and intractable. We
formulate a particular convex maximization, here called the pointwise dual
relaxation (PDR), whose supremum is a lower bound on the infimum of the
original problem. The PDR can be derived by dualizing and relaxing the original
problem; its constraints are pointwise equalities or inequalities over
finite-dimensional sets, rather than over infinite-dimensional function spaces.
When the original minimization can be specified by polynomial functions of
$(x,u,\nabla u)$, the PDR can be further relaxed by replacing pointwise
inequalities with polynomial sum-of-squares (SOS) conditions. The resulting SOS
program is computationally tractable when the dimensions $m,n$ and number of
constraints are not too large. The framework presented here generalizes an
approach of Valmorbida, Ahmadi, and Papachristodoulou (IEEE Trans. Automat.
Contr., 61:1649--1654, 2016). We prove that the optimal lower bound given by
the PDR is sharp for several classes of problems, whose special cases include
leading eigenvalues of Sturm-Liouville problems and optimal constants of
Poincar\'e inequalities. For these same classes, we prove that SOS relaxations
of the PDR converge to the sharp lower bound as polynomial degrees are
increased. Convergence of SOS computations in practice is illustrated for
several examples.
