I guess the title kind of speaks for my questions: I'm curious to know what could be the physical interpretation or real life application of the concept of regularity that arises in PDE: take for example, the Laplace equation: assume $u$ is continuous upto the boundary of the unit disk $\mathbb{D}$.

$\Delta u=0$ in the unit disk $\mathbb{D}$, $u=g$ on $\partial\mathbb{D}$. Now if $g\in C^{k, \alpha}(\partial\mathbb{D})$, then $u\in C^{k+1, \alpha}(\mathbb {\bar{D}})$.

The above is an example of boundary regularity: but what it really means in physical/engineering terms?

I guess I'm looking for a more general answers to the physical/real life interpretations of interior r boundary regularity of solutions to PDE's.

Thanks!