lower semicontinuity

When you take the limit of a minimising sequence, the one thing you cannot afford is for energy to mysteriously vanish in the limit — for the limit function to be cheaper than anything in the sequence allowed. Lower semicontinuity is the precise property that forbids this downward jump. It is a one-sided weakening of continuity: the function may jump UP in the limit, but it can never jump DOWN.

A functional E is lower semicontinuous along a sequence if, whenever u_k converges to u, the value at the limit is no larger than the limiting values: E[u] is not greater than the liminf of E[u_k]. Continuity would demand equality, E[u] = lim E[u_k]; lower semicontinuity asks only for the inequality in the safe direction. The reason this weaker condition is exactly what variational existence needs: in the direct method the convergence available is only WEAK (weak compactness is all coercivity buys you), and energy functionals are typically NOT weakly continuous — a rapidly oscillating sequence can carry energy that the smooth weak limit does not see, so energy genuinely leaks. But it leaks DOWNWARD only: the limit's energy can dip below the limiting energies, never rise above. Forbidding exactly that downward leak is lower semicontinuity, and it is the most that is true and all that is needed.

The workhorse criterion is convexity. For an energy E[u] = the integral of L(x, u, grad u), the functional is weakly lower semicontinuous provided the Lagrangian L is convex in the gradient variable (Tonelli's theorem); for vector-valued problems convexity is relaxed to the subtler quasiconvexity of Morrey. This is why convex energies — the Dirichlet energy, p-energies, the Lagrangian of elasticity in the convex regime — are so well behaved, and why the calculus of variations devotes so much care to convexity. Lower semicontinuity is the certificate that the candidate caught by coercivity is genuinely the minimiser.