When does a weak solution give rise to a strong solution?

Question

When does a weak solution of a PDE give rise to an actual solution?

I know that this is true for the Laplace equation thanks to Weyl's lemma. Does this happen in other cases?

More generally what can be deduced from the existence of a weak solution?

Answer 1

When you have a weak problem of the form $$ \text{find}\; u \in U: \quad a(u,v) = (f,v), \quad \forall v \in V, $$ arising from a strong formulation $L u = f$, the regularity of the solution $u$ usually depends on that of $f$ and on the order of the differential operator $L$.

Intuitively, for example, if $L= \Delta$, you want $u\in H^2$ in order to fulfill the strong formulation. A sufficient condition for this is $f \in L^2$ (see Theorem 2 in these notes).