For p>1, the relative p-Faber-Krahn inequality is shown to be equivalent to volume doubling together with a sub-Gaussian upper estimate for Trudinger subsolutions, with improved long-time bounds under uniform Faber-Krahn decay.
Doubling measures, Poincar\'e inequalities and parabolic Harnack inequalities for a doubly nonlinear equation
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We characterize metric measure spaces satisfying parabolic Harnack inequalities for a doubly nonlinear equation in terms of volume doubling and Poincar\'e inequalities. Our approach uses purely analytical methods, based on obtaining estimates for solutions to a related Cauchy problem. This extends previous linear results to the nonlinear setting without relying on heat kernel estimates and representation formulae.
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Relative Faber--Krahn inequalities and Trudinger's equation on Riemannian Manifolds
For p>1, the relative p-Faber-Krahn inequality is shown to be equivalent to volume doubling together with a sub-Gaussian upper estimate for Trudinger subsolutions, with improved long-time bounds under uniform Faber-Krahn decay.