It is known that solutions to second order uniformly elliptic and parabolic equations, either in divergence or nondivergence (general) form, are Hölder continuous and satisfy the interior Harnack inequality. We show that even in the one-dimensional case (x ∈ R1), these properties are not preserved for equations of mixed divergence-nondivergence structure: for elliptic equations Di(a 1 ijDju) + ...

In this paper, we establish new generalizations and results in shift-invariant subspaces of mixed-norm Lebesgue spaces Lp→(Rd). We obtain a Hölder inequality, Minkowski convolution convolution-Hölder type inequality stability theorem to case the setting subspace Our unify refine existing literature.

We obtain a local weighted Caccioppoli-type estimate and prove the weighted version of the weak reverse Hölder inequality for A-harmonic tensors.

We study partial regularity of very weak solutions to some nonhomogeneous A-harmonic systems. To obtain the reverse Hölder inequality of the gradient of a very weak solution, we construct a suitable test function by Hodge decomposition. With the aid of Gehring's lemma, we prove that these very weak solutions are weak solutions. Further, we show that these solutions are in fact optimal Hölder co...

This paper is concerned with existence of a C viscosity solution of a second order nontranslation invariant integro-PDE. We first obtain a weak Harnack inequality for such integroPDE. We then use the weak Harnack inequality to prove Hölder regularity and existence of solutions of the integro-PDEs.

We obtain a Harnack type inequality for solutions to elliptic equations in divergence form with non-standard p(x)-type growth. A model equation is the inhomogeneous p(x)-Laplacian. Namely, ∆p(x)u := div ( |∇u|p(x)−2∇u ) = f(x) in Ω, for which we prove Harnack’s inequality when f ∈ Lq0 (Ω) if max{1, N p1 } < q0 ≤ ∞. The constant in Harnack’s inequality depends on u only through ‖|u|p(x)‖p2−p1 L1...