Maximum principles for the relativistic heat equation

Evan Miller; Ari Stern

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arXiv · 1507.05030

Maximum principles for the relativistic heat equation

Evan Miller·Ari Stern

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Abstract

The classical heat equation is incompatible with relativity, since the strong maximum principle allows for disturbances to propagate instantaneously. Some authors have proposed limiting the propagation speed by adding a linear hyperbolic correction term, but then even a weak maximum principle fails to hold. We study a more recently introduced relativistic heat equation, which replaces the Laplace operator by a quasilinear elliptic operator, and show that strong and weak maximum principles hold for stationary and time-varying solutions, respectively, as well as for sub- and supersolutions. Moreover, by transforming the equation into an equivalent form, related to the mean curvature operator, we prove even stronger tangency and comparison principles.

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Evan Miller, Ari Stern. 2015-07-17. Maximum principles for the relativistic heat equation. https://arxiv.org/abs/1507.05030

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