Prove inequality in Sobolev space 1

by w.sdka   Last Updated October 17, 2018 16:20 PM

$\textbf{Problem}$ Integrate by parts to prove: \begin{align*} \int_{\Omega} \vert Du \vert^p dx \leq C\Vert u \Vert_{L^p(\Omega)}^{1/2}\Vert D^2u\Vert_{L^p(\Omega)}^{1/2} \end{align*} for $2\leq p <\infty $ and all $u\in W^{2,p}(\Omega) \cap W_0^{1,p}(\Omega) $.

$\textbf{Notation}$ (i) A vector of the form $\alpha = (\alpha_1,\cdots,\alpha_n)$, where each component $\alpha_i$ is a nonnegative integer, is called a multiindex of order \begin{align*} \vert \alpha \vert =\alpha_1+\cdots+\alpha_n \end{align*} (ii) Given a multiindex $\alpha$, define \begin{align*} D^{\alpha}u(x):=\frac{\partial^{\vert \alpha \vert}u(x)}{\partial_{x_1}^{\alpha_1}\cdots\partial_{x_n}^{\alpha_n}} \end{align*} (iii) If $k$ is a nonnegative integer, \begin{align*} D^ku(x):= \{D^{\alpha}u(x) \vert \; \vert \alpha \vert =k\} \end{align*} the set of all partial derivatives of order $k$.

(iv) $\vert D^ku \vert = (\sum_{\vert \alpha \vert =k } \vert D^{\alpha}u\vert^2)^{1/2}$

$\textbf{Attempt} $ \begin{align*} \int_{\Omega} \vert Du \vert ^p \; dx &=\int_{\Omega} \nabla u \cdot \nabla u \vert Du \vert ^{p-2} \; dx \\ &=\int_{\Omega} \nabla \cdot(u \vert Du \vert^{p-2} \nabla u)\; dx - \int_{\Omega} u\nabla \cdot (\vert Du \vert^{p-2} \nabla u)\; dx\\ &=\int_{\partial \Omega} (u\vert Du \vert^{p-2})\cdot n \; d\sigma -\int_{\Omega} u\nabla \cdot (\vert Du \vert^{p-2} \nabla u)\; dx \quad \because \textrm{ divergence theorem}\\ &=-\int_{\Omega} u\nabla \cdot (\vert Du \vert^{p-2} \nabla u)\; dx \quad \because u \in W_0^{1,p}(\Omega)\\ &=-\int_{\Omega} u(\Delta u \vert Du \vert^{p-2} + (p-2) (\nabla u^{T}D^2u\nabla u)\vert Du \vert^{p-4}) \; dx \\ &=-\int_{\Omega} u\vert Du \vert^{p-2}(\Delta u + (p-2)(\nabla u^{T}D^2u\nabla u)\vert Du \vert^{-2}) \; dx\\ &\leq C \int_{\Omega} u \vert Du \vert^{p-2} \vert D^2u \vert \; dx \quad (?) \end{align*}

I don't know how to find the constant $C$ such that \begin{align*} \Delta u + (p-2)(\nabla u^{T}D^2u\nabla u)\vert Du \vert^{-2} \leq C \vert D^2u \vert \end{align*} Any help is appreiciated...

Thank you!



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