Regularity of minimizers

Remarks

Here $C^{k,\alpha}$ is the space of $C^k$ maps whose $k$-th derivatives are locally Hölder continuous with exponent $\alpha$.

Example: for $m = 1$ and $L(\xi) = \frac{1}{2} \abs{\xi}^2$, minimisers are weak solutions of $\Delta u = 0$. By Weyl's lemma they are harmonic, hence smooth, so they are regular for every $k$ and $\alpha$.

Non-example: for $n = 1$, $m = 1$ and $L(x, u, \xi) = (\xi^2 - 1)^2$ on $\Omega = (0, 2)$, the function $u(x) = \min(x, 2 - x)$ is a minimiser among maps with its boundary values, since $J(u) = 0$. It is Lipschitz but not $C^1$, so it is not regular in the sense of clause 1. Here $L$ is not convex in $\xi$.

For $m = 1$ and smooth, uniformly convex $L(\xi)$ with bounded second derivatives, every minimiser in $W^{1,2}$ is smooth, by the De Giorgi–Nash–Moser theorem and Schauder estimates. For $m \ge 2$ this fails in general, and partial regularity is the typical result for smooth, uniformly convex or strongly quasiconvex integrands with suitable growth.