*Submitted Paper*

**Inserted:** 10 feb 2020

**Last Updated:** 17 may 2020

**Year:** 2020

**Abstract:**

For $q>2, \gamma > 1$, we prove that maximal regularity of $L^q$ type holds for periodic solutions to
$-\Delta u +

Du

^{\gamma} = f$ in $\mathbb{R}^d$, under the (sharp) assumption $q > d \frac{\gamma-1}\gamma$.

**Keywords:**
maximal regularity, Kardar-Parisi-Zhang equation, Riccati equation, Bernstein method

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