Higher integrability for parabolic PDEs with generalized Orlicz growth
A talk in the BI.discrete series by
Jihoon Ok from Sogang University Seoul
| Abstract: | In this talk, I will present a higher integrability result of the gradient of weak solutions to the parabolic $A$-system whose prototype is
$$
\partial_t u - \mathrm{div}\!\left(\frac{\varphi'(z, |\nabla u|)}{|\nabla u|}\nabla u\right) = 0, \qquad u = (u^1, \dots, u^N),
$$
where $\varphi$ is a generalized Young function.
Our main result includes previously known results for the $p$-growth, the variable exponent and the double phase growth as special cases. Also included are previously unknown borderline double phase growth and perturbed variable exponent growth, among others.
The problem is controlled by a natural requirement of comparison of $\varphi$ between points in intrinsic parabolic cylinders via an (A1)-condition, which unifies disparate conditions from the special cases. Moreover, we handle both the singular and degenerate cases at the same time, providing a simple proof of a reverse Hölder type inequality, which is new even in the $p$-growth case. This is a joint work with Peter Hästö from University of Helsinki. Within the CRC this talk is associated to the project(s): A7 |