Interior and boundary higher integrability of very weak solutions for quasilinear parabolic equations with variable exponents. (May 2020) - Adimurthi, Karthik; Byun, Sun-Sig; Oh, Jehan
Abstract: We prove boundary higher integrability for the (spatial) gradient of very weak solutions of quasilinear parabolic equations of the form u t − div A ( x, t, ∇ u ) = 0 on Ω × ( − T, T ), u = 0 on ∂ Ω × ( − T, T ), where the non-linear structure A ( x, t, ∇ u ) is modeled after the variable exponent p ( x, t ) -Laplace operator given by | ∇ u | p ( x, t ) − 2 ∇ u . To this end, we prove that the gradients satisfy a reverse Hölder inequality near the boundary by constructing a suitable test function which is Lipschitz continuous and preserves the boundary values. In the interior case, such a result was proved in Verena Bögelein and Qifan Li (2014) provided p ( x, t ) ≥ p − ≥ 2 holds and was then extended to the singular case 2 n n + 2 < p − ≤ p ( x, t ) ≤ p + ≤ 2 in Qifan Li (2017). This restriction was necessary because the intrinsic scalings for quasilinear parabolic problems are different in the case p + ≤ 2 and p − ≥ 2 . In this paper, we develop a new approach, using which, we are able to extend the results of Verena Bögelein and Qifan Li (2014), Qifan Li (2017) to the full range 2 n n + 2 < p − ≤ p ( x, t ) ≤ p + < ∞ and also obtain analogous results up to the boundary. The main novelty of this paper is that we make use of a unified intrinsic scaling using which our methods are able to handle both the singular case and degenerate case simultaneously . Our techniques improve and simplify many aspects of the method of parabolic Lipschitz truncation (even in the constant exponent case) studied extensively in existing literature. To simplify the exposition, we will only prove the higher integrability result near the boundary, provided the domain Ω satisfies a uniform measure density condition and are non perturbative in nature, hence we make no regularity assumptions for the coefficients of the nonlinear operator. Our techniques are also applicable to higher order equations as well as systems.