Harnack’s estimate for a mixed local–nonlocal doubly nonlinear parabolic equation

Wolff potentials and measure data vectorial problems with Orlicz growth

“…Note that we also have (51). Thus, we can estimate the first and second integrals on the right-hand side by using Lemmas 4.3 and 4.2, respectively, which leads to the conclusion.…”

“…Studies on regularity results for mixed local-nonlocal p-Laplacian type operators were initiated in the paper [36], where the De Giorgi-Nash-Moser theory was investigated by combining the techniques for the fractional p-Laplacian [28,29] with those for the classical p-Laplacian. We refer to [2,11,14,37,38,51] and references therein for further results concerning mixed local-nonlocal nonlinear problems. We would like to single out the paper [27] where maximal regularity, including (local) gradient Hölder regularity, was proved for a larger class of mixed problems related to −∆ p + (−∆ γ ) s with p, γ > 1 and s ∈ (0, 1) satisfying p > sγ.…”

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

“…Remark 2.17. We remark that the definition of weak sense in Theorem 2.15 and Theorem 2.16 is analogous to the doubly nonlinear equation below in subsection 3.2 and the admissibility of the test function φ = max{u − v, 0} in Theorem 2.15 and Theorem 2.16 can be justified using the mollification technique, for example in [24,57].…”

“…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.…”

“…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.…”