Higher integrability for parabolic double phase equations with an improved gap bound

Abstract

We prove a local higher integrability result for the gradient of Hölder continuous weak solutions to the parabolic double phase equation \[ \partial_t u - \operatorname{div} \left(|Du|^{p-2}Du + a(z)|Du|^{q-2}Du\right) = 0 \qquad \text{in } Ω_T. \] We work under a relaxed gap condition on the exponents $p$ and $q$. The coefficient $a$ is assumed to belong to the class ${\mathcal{Z}}^{\hat{I}}º(\hat{I}{\text{c◯}}_T)$ for some $\hat{I}º\in (0,\infty )$. The functions in this class satisfy a one-sided pointwise bound that controls how fast $a$ can grow away from its zero set, and the class contains the Hölder continuous functions. We also impose a mild almost increasing condition on $a$, which motivates the introduction of a new mollification, which we call the slanted Steklov average. For $u\in C_{\mathrm{loc}}^{0,{\hat{I}}^3,{\hat{I}}^3/q}(\hat{I}{\text{c◯}}_T)$ with ${\hat{I}}^3\in [0,1)$, our main result holds under the gap bound \begin{equation}\tag{G}\label{eq:G} 2 \le p \le q \le p + \frac{qκ}{q - 2γ}. \end{equation} The new gap condition \eqref{eq:G} is purely parabolic in nature and is stricter than the optimal gap relation associated with the Lavrentiev phenomenon for the elliptic double phase functional.

40 pages, comments are welcome!