We study a parabolic equation for the fractional p-Laplacian of order s, for $$p\ge 2$$
p
≥
2
and $$0<s<1$$
0
<
s
<
1
. We provide space-time Hölder estimates for weak solutions, with explicit exponents. The proofs are based on iterated discrete differentiation of the equation in the spirit of Moser’s technique.
We state and prove estimates for the local boundedness of subsolutions of non-local, possibly degenerate, parabolic integro-differential equations of the formwhere P.V. means in the principle value sense, p ∈ (1, ∞) and the kernel obeys K(x, y, t) ≈ |x − y| n+ps for some s ∈ (0, 1), uniformly in (x, y, t) ∈ R n × R n × R.
The main result of this paper is a nonlocal version of Harnack's inequality for a class of parabolic nonlocal equations. We additionally establish a weak Harnack inequality as well as local boundedness of solutions. None of the results require the solution to be globally positive.2010 Mathematics Subject Classification. 35K10, 35B65, 35R11.
Abstract. In this paper we study the homogenization of p−Laplacian with thin obstacle in a perforated domain. The obstacle is defined on the intersection between a hyperplane and a periodic perforation.We construct the family of correctors for this problem and show that the solutions for the ε−problem converge to a solution of a minimization problem of similar form but with an extra term involving the mean capacity of the obstacle. The novelty of our approach is based on the employment of quasi-uniform convergence. As an application we obtain Poincaré's inequality for perforated domains.
Abstract. We prove the existence of big pieces of regular parabolic Lipschitz graphs for a class of parabolic uniform rectifiable sets satisfying what we call a synchronized two cube condition. An application to the fine properties of parabolic measure is given.2000 Mathematics Subject Classification.
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