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In this note, we consider regularity theory for a fractional p-Laplace operator which arises in the complex interpolation of the Sobolev spaces, the Hs,p-Laplacian. We obtain the natural analogue to the classical p-Laplacian situation, namely Cs+αloc-regularity for the homogeneous equation.

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We establish the variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in temporally random shear flows with sufficiently decaying correlations. A key quantity in the variational principle is the almost sure Lyapunov exponent of a heat operator with random potential. To prove the variational principle, we use the comparison principle of solutions, the path integral representation of solutions, and large deviation estimates of the associated stochastic flows. The variational principle then allows us to analytically bound the front speeds. The speed bounds imply the linear growth law in the regime of large root mean square shear amplitude at any fixed temporal correlation length, and the sublinear growth law if the temporal decorrelation is also large enough, the so-called bending phenomenon.

We establish the solvability of second order divergence-type parabolic systems in Sobolev spaces. The leading coefficients are assumed to be merely measurable in one spatial direction on each small parabolic cylinder with the spatial direction allowed to depend on the cylinder. In the other orthogonal directions and the time variable, the coefficients have locally small mean oscillations. We also obtain the corresponding $W^1_p$-solvability of second order elliptic systems in divergence form. This type of system arises from the problems of linearly elastic laminates and composite materials. Our results are new even for scalar equations, and the proofs differ from and simplify the methods used previously in [H. Dong and D. Kim, <i>Arch. Ration. Mech. Anal.</i>, 196 (2010), pp. 2570]. As an application, we improve a result by Chipot, Kinderlehrer, and Vergara-Caffarelli [<i>Arch. Ration. Mech. Anal.</i>, 96 (1986), pp. 8196] on