Abstract:We extend the method of modulus of continuity for solutions of parabolic equations--as used, for instance, to prove the Fundamental Gap Conjecture--to solutions of non-local heat equations on R^n and in dimension one with a non-local Neumann boundary condition. Specifically, we show that if a solution of a non-local heat equation has an initial modulus of continuity satisfying simple criteria, then this modulus of continuity is preserved at all subsequent times. In the process of trying to generalise our result in one dimension, we found a counterexample suggesting that a non-local analogue of the Payne-Weinberger inequality would depend on more than the diameter of a bounded (convex) domain.
From: Sophie Chen [view email]
[v1]
Fri, 25 Jul 2025 07:22:12 UTC (172 KB)
[v2]
Thu, 13 Nov 2025 03:23:04 UTC (173 KB)
[v3]
Wed, 10 Jun 2026 03:04:33 UTC (183 KB)
[v4]
Fri, 10 Jul 2026 01:08:47 UTC (183 KB)
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