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Minor additions in order to (hopefully) clarify.
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Daniele Tampieri
Daniele Tampieri
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My two cents:. Despite being a bit far away from the main theme of the question (statistics and optimization), from the point of view of PDEs I think that a particularly illuminating reference on the importance of Lipschitz/Hölder functions is the recent book of Fiorenza [1]: in. In this context, the Lipschitz/Hölder continuity requirement is the classical means to control e.g. the behavior of the inner normal vector to a Lyapunov manifold (i.e. a manifold whose representing function is locally Hölder continuous), obtain several classical results on the solvability of boundary value problems for elliptic equations and analyzing the behavior of potentials across a (hyper)surface discontinuity (the classical Plemelj formula is proved assuming Lipschtz continuity of the boundary value of the given harmonic function).

Reference

[1] Renato Fiorenza, Hölder and locally Hölder continuous functions, and open sets of class $C^k$, $C^{k,\lambda}$ (English), Frontiers in Mathematics, Basel: Birkhäuser/Springer (ISBN 978-3-319-47939-2/pbk; 978-3-319-47940-8/ebook), pp. xi+152 (2016), MR3588287, Zbl 1366.26003.

My two cents: from the point of view of PDEs I think that a particularly illuminating reference is the recent book of Fiorenza [1]: in this context, the Lipschitz/Hölder continuity requirement is the classical means to control e.g. the behavior of the inner normal vector to a Lyapunov manifold (i.e. a manifold whose representing function is locally Hölder continuous), obtain several classical results on the solvability of boundary value problems for elliptic equations and analyzing the behavior of potentials across a (hyper)surface discontinuity (the classical Plemelj formula is proved assuming Lipschtz continuity of the boundary value of the given harmonic function).

Reference

[1] Renato Fiorenza, Hölder and locally Hölder continuous functions, and open sets of class $C^k$, $C^{k,\lambda}$ (English), Frontiers in Mathematics, Basel: Birkhäuser/Springer (ISBN 978-3-319-47939-2/pbk; 978-3-319-47940-8/ebook), pp. xi+152 (2016), MR3588287, Zbl 1366.26003.

My two cents. Despite being a bit far away from the main theme of the question (statistics and optimization), from the point of view of PDEs I think that a particularly illuminating reference on the importance of Lipschitz/Hölder functions is the recent book of Fiorenza [1]. In this context, the Lipschitz/Hölder continuity requirement is the classical means to control e.g. the behavior of the inner normal vector to a Lyapunov manifold (i.e. a manifold whose representing function is locally Hölder continuous), obtain several classical results on the solvability of boundary value problems for elliptic equations and analyzing the behavior of potentials across a (hyper)surface discontinuity (the classical Plemelj formula is proved assuming Lipschtz continuity of the boundary value of the given harmonic function).

Reference

[1] Renato Fiorenza, Hölder and locally Hölder continuous functions, and open sets of class $C^k$, $C^{k,\lambda}$ (English), Frontiers in Mathematics, Basel: Birkhäuser/Springer (ISBN 978-3-319-47939-2/pbk; 978-3-319-47940-8/ebook), pp. xi+152 (2016), MR3588287, Zbl 1366.26003.

Source Link
Daniele Tampieri
Daniele Tampieri
  • 6.4k
  • 7
  • 30
  • 45

My two cents: from the point of view of PDEs I think that a particularly illuminating reference is the recent book of Fiorenza [1]: in this context, the Lipschitz/Hölder continuity requirement is the classical means to control e.g. the behavior of the inner normal vector to a Lyapunov manifold (i.e. a manifold whose representing function is locally Hölder continuous), obtain several classical results on the solvability of boundary value problems for elliptic equations and analyzing the behavior of potentials across a (hyper)surface discontinuity (the classical Plemelj formula is proved assuming Lipschtz continuity of the boundary value of the given harmonic function).

Reference

[1] Renato Fiorenza, Hölder and locally Hölder continuous functions, and open sets of class $C^k$, $C^{k,\lambda}$ (English), Frontiers in Mathematics, Basel: Birkhäuser/Springer (ISBN 978-3-319-47939-2/pbk; 978-3-319-47940-8/ebook), pp. xi+152 (2016), MR3588287, Zbl 1366.26003.