# Integrability condition on function determining PDE domain

I'm currently looking through the following paper which examines some dynamics of the Airy$$_2$$ process: https://arxiv.org/pdf/1106.2717.pdf

On page 2, there appears a PDE of the form

$$\partial_t u + Hu = 0, \quad x \leq g(t)$$

with boundary conditions

$$u(t,x) = 0, \quad x > g(t)$$,

$$u(l, x) = f(x)$$ if $$x < g(t)$$

where $$H = - \partial_x^2 + x$$ is the Airy Hamiltonian.

Now, this means that $$g$$ describes the domain in space - however I don't understand why one might require it to be in $$H^1$$. The authors point to the proof of Proposition 3.2, but there only a generalisation from a particular solution to the $$g\in H^1$$ case is done - nothing indicates that that integrability should be necessary for the result to hold.

$$H^1$$ in one dimension doesn't seem sufficient to get a lot of regularity - if I'm not mistaken, we get that $$g\in L^\infty \cap C^{0,\frac{1}{2}}$$, but nothing like being locally Lipschitz or other conditions you might want in connection with a PDE that would allow one to, for example, apply divergence theorem.

So, are there situations where this kind of $$H^1$$ regularity on a domain is important, or was it just some class for which the proof was done in this paper.

An additional doubt I have is that in context, the statement should work even if just $$g(l), \ g(r)$$ are prescribed, and $$g(t) = \infty$$ everywhere else, and obviously that lacks any integrability or differentiability whatsoever.