Abstract
The objective of this work is an existence proof for variational solutions u to parabolic minimizing problems. Here, the functions being considered are defined on a metric measure space ({mathcal {X}}, d, mu ). For such parabolic minimizers that coincide with Cauchy-Dirichlet data eta on the parabolic boundary of a space-time-cylinder varOmega times (0, T) with an open subset varOmega subset {mathcal {X}} and T > 0, we prove existence in the parabolic Newtonian space L^p(0, T; {mathcal {N}}^{1,p}(varOmega )). In this paper we generalize results from Collins and Herán (Nonlinear Anal 176:56–83, 2018) where only time-independent Cauchy–Dirichlet data have been considered. We argue completely on a variational level.
Highlights
The aim of this paper is to show existence for parabolic minimizers on metric measure spaces
The idea of considering variational problems on metric measure spaces is based on independent proofs of Grigor’yan [23] and Saloff-Coste [51] of the fact that on Riemannian manifolds the doubling property and Poincaré inequality are equivalent to a certain Harnack-type inequality for solutions of the heat equation
In addition to the doubling property, we demand that the metric measure space (X, d, ) supports a weak (1, p)-Poincaré inequality, in the sense that there exist a constant cP > 0 and a dilatation factor ≥ 1 such that for all open balls Bρ(x0) ⊂ Bτρ(x0) ⊂ X, for all Lp -functions u on X and all upper gradients gu of u there holds p
Summary
The aim of this paper is to show existence for parabolic minimizers on metric measure spaces. The idea of considering variational problems on metric measure spaces is based on independent proofs of Grigor’yan [23] and Saloff-Coste [51] of the fact that on Riemannian manifolds the doubling property and Poincaré inequality are equivalent to a certain Harnack-type inequality for solutions of the heat equation. Existence for parabolic problems on metric measure spaces has already been dealt with in [13] with boundary data independent from time. If we are going to generalize results that have been found for integrands f ∶ Rn → R in the Euclidean case, we have to think about what happens if the integrand only depends on the modulus, i.e. f ( ) = g(| |) for a convex function g. To obtain the existence result for more regular data, we introduce a mollification in time
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