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What are the examples of practical applications of Sobolev spaces?

The framework of Sobolev spaces is very useful in the theoretical analysis of PDEs and variational problems: the questions of existence, stability and regularity of solutions, convergence of numerical methods etc. But what are the examples of less theoretical uses of Sobolev spaces? In particular, what are the examples of

  • PDEs whose explicit solution can only be found using the theory of Sobolev spaces
  • problems when Sobolev spaces are an essential ingredient of otherwise impossible computation of some physically relevant quantity (heat flux, drag force etc.)
  • PDEs from Physics (or other areas) when the solutions are not smooth but are Sobolev (by nature of the problem).

Remark. Maybe this has to be a community wiki question.

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Formulating optimization/control problems in Sobolev spaces often lead to better numerically conditioned problems, and more practically implementable solutions.

E.g: Consider the problem of devising efficient fluid mixing techniques:

Foures, D. P. G., C. P. Caulfield, and Peter J. Schmid. "Optimal mixing in two-dimensional plane Poiseuille flow at finite Péclet number." Journal of Fluid Mechanics 748 (2014): 241-277.

Available at https://core.ac.uk/download/pdf/52899232.pdf

The core idea being that rather than trying to mix at all (fourier/wavelength) scales equally well , it is better to formulate the cost function that penalizes large scale inhomogenities more than fine variations in concentration.

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What about the Eikonal equation $$ |\nabla u| = 1? $$ As far as I understand, one can construct solutions directly, but still, the solution does not have the desired classical regularity in general but only weak solutions with regularity in some Sobolev space.

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