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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!

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  • $\begingroup$ Just an idea: Via Fourier transform and other such transforms higher regularity corresponds to faster decay at infinity meaning that higher frequency parts of your function occur with smaller amplitude. That sounds like a real-world property of the function in question. (Although I'm very far from being a physicist) $\endgroup$
    – Johannes Hahn
    Commented Apr 20, 2014 at 21:02

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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.

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    $\begingroup$ good example. a similar example also works for the wave or plate equation, which is seemingly one of the reasons why windows on planes have round corners (i.e., regularity of the boundary avoids cracks). $\endgroup$
    – Delio Mugnolo
    Commented Apr 21, 2014 at 15:54
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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.

  1. Continuity of position means no jumps in space (i.e. no teleportation);
  2. Continuity of the velocity (first derivative of position) means no sudden moves;
  3. Continuity of the acceleration (second derivative of position) means no impact forces;
  4. Continuity of the jog (third derivative of position) means smooth force changes.

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

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