Physical and real life interpretation of the concept of regularity used in differential equations? 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!
 A: There is a physical phenomenon where lack of boundary regularity becomes evident. If you take a metal cone and put it at a sufficiently high electric potential, then the electric field at the tip becomes very strong, often enough to ionize the air. A real life consequence of this is that lightnings are more likely to hit pointed metal objects. This phenomenon happens because you are trying to solve the Dirichlet problem on a domain with an irregular boundary (the complement of the cone), for which the elliptic regularity estimates don't hold up to the boundary.  
A: An obvious other example is that of shock waves, which arise as discontinuous solutions of hyperbolic equations; in the case of acoustic waves in air, we hear the shock as a bang.
More generally, even for ODEs in daily life we can easily notice various levels of regularity. Take the example of a rider in a carnival ride. 


*

*Continuity of position means no jumps in space (i.e. no teleportation);

*Continuity of the velocity (first derivative of position) means no sudden moves;

*Continuity of the acceleration (second derivative of position) means no impact forces;

*Continuity of the jog (third derivative of position) means smooth force changes.


The higher the regularity, the smoother the ride ...
