We prove local boundedness of variational solutions to the double phase equationunder the restrictions s, s ′ ∈ (0, 1), 2N 2s + N ≤ p ≤ q ≤ p 2s + N N and the function (x, y, t) → a(x, y, t) is assumed to be measurable and bounded.

We prove existence of variational solutions for a class of nonlocal evolution equations whose prototype is the double phase equationThe approach of minimization of parameter-dependent convex functionals over space-time trajectories requires only appropriate convexity and coercivity assumptions on the nonlocal operator. As the parameter tends to zero, we recover variational solutions. Under further growth conditions, these variational solutions are global weak solutions. Further, this provides a direct minimization approach to approximation of nonlocal evolution equations.

We prove local Hölder regularity for a nonlocal parabolic equations of the formfor p ∈ (1, ∞) and s ∈ (0, 1).

We give an alternative proof for Hölder regularity for weak solutions of nonlocal elliptic quasilinear equations modelled on the fractional p-Laplacian where we replace the discrete De Giorgi iteration on a sequence of concentric balls by a continuous iteration. Moreover, we also obtain a De Giorgi type isoperimetric inequality for all s ∈ (0, 1), which partially answers a question of M.Cozzi.

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