Elliptic regularity

Remarks

Proof idea for $m = 0$: multiply the weak form by a cut-off function and test it with difference quotients $D_k^h u$ of $u$ in the direction $e_k$. Uniform ellipticity bounds $\norm{D_k^h \nabla u}_{L^2}$ on $V$ independently of $h$, and a uniform bound on difference quotients implies that the weak derivative $\partial_k \nabla u$ exists in $L^2(V)$. The case of general $m$ follows by induction, since a derivative of $u$ solves an equation of the same form with a right-hand side in $H^{m-1}$. Part 2 follows from part 1 for all $m$ together with the Sobolev embedding theorem, which puts $H^{k}$ inside $C^j$ for $k > j + n/2$.

Example: a weak solution $u \in H^1(U)$ of $\Delta u = 0$ is smooth; this is Weyl's lemma, and $u$ is then a classical harmonic function.

Ellipticity is needed. For the wave operator $\partial_t^2 - \partial_x^2$ in the plane, $u(t, x) = \abs{x - t}$ is a weak solution of $\partial_t^2 u - \partial_x^2 u = 0$ that is not differentiable on the line $x = t$.

The regularity is local: without conditions on the boundary of $U$ and on the boundary values, $u$ need not be in $H^{m+2}(U)$ up to the boundary. For the Dirichlet problem with $u \in H^1_0(U)$ and $\partial U$ of class $C^{m+2}$, the same conclusion holds on all of $U$.

Going deeper

For a student

The standard way to solve a linear elliptic problem has two steps. First one gets a weak solution in $H^1$ from the Lax–Milgram theorem or the Riesz representation theorem. Then elliptic regularity upgrades it to $H^{m+2}$, and with Sobolev embeddings to a classical solution. The same pattern holds in other function spaces: the Calderón–Zygmund estimates give the $W^{2,p}$ version for $1 < p < \infty$, and the Schauder estimates give the $C^{2,\alpha}$ version. Hyperbolic operators have no such property, since their solutions propagate singularities of the data.