Local boundedness of a mixed local–nonlocal doubly nonlinear equation

Abstract

We establish Harnack's estimates for positive weak solutions to a mixed local and nonlocal doubly nonlinear parabolic equation, of the typewhere the vector field A satisfies the p-ellipticity and growth conditions and the integrodifferential operator L whose model is the fractional p-Laplacian. All results presented in this paper are provided together with quantitative estimates.

Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems

“…The doubly nonlinear mixed equation (3.44) is recently studied. Nakamura [56,57] proved local boundedness and Harnack inequality for (3.44) assuming that α = p − 1 and g ≡ 0. The case α > 0 is recently studied in [60] to obtain Harnack inequality for the equation (3.44) assuming that H(x) = |x| and g ≡ 0.…”

“…Below, we establish energy estimates for subsolutions and supersolutions. In the mixed case, see [41,56,57] and the references therein. First, we prove the following energy estimate for weak supersolutions of (3.44).…”

“…To the best of our knowledge, most of our results are valid even for p = 2, that is for the operator −∆u + (−∆) s . Regarding the anisotropic mixed equation, as far we are aware, some regularity results are recently developed in [42,39,56,57] and also related eigenvalue and extremal problems are studied, see [38,45]. We also refer to [4,40] and the references therein.…”