Problem 9 - Chapter 5 - Evans' PDE (First Edition)

As you guessed, the first step is an integration by parts: $$\int_U |Du|^p dx = \int_U \nabla u \cdot \nabla u |Du|^{p-2} dx = - \int_U u \nabla \cdot ( \nabla u |Du|^{p-2}) dx = - \int_U u \left( \Delta u |Du|^{p-2} + (p-2) (\nabla u^T D^2 u \nabla u) |Du|^{p-4})\right) \leq C \int_U u |Du|^{p-2} |D^2 u| dx $$ Now the next step is an invocation of the Holder inequality, with conjugate exponents $\frac{p}{2}$ and $\frac{p}{p-2}$ (notice that this step requires $p \gt 2$, if $p=2$, it is unnecessary): $$\int_U u |Du|^{p-2} |D^2 u| dx \leq \left(\int_U |u|^\frac{p}{2} |D^2u|^\frac{p}{2}\right)^\frac{2}{p} \left(\int_U |Du|^p \right)^\frac{p-2}{p} $$ So, dividing the original left hand side by the gradient term and invoking Holder on the remaining term with exponent 2, we get the desired result. $$ \int_U |Du|^p dx \leq C \left(\int_U |u|^p dx \right)^\frac{1}{2} \left( \int_U |D^2u|^p dx \right)^\frac{1}{2} $$