I have a question regarding the proof of Reshetnyak's lemma.

Reshetnyak: Let $\Omega \subseteq \mathbb{R}^2$ open, bounded with Lipschitz boundary. Let $p>2$ and $(u_k)_k\subseteq W^{1,p}(\Omega,\mathbb{R}^2)$, $u\in W^{1,p}(\Omega,\mathbb{R}^2)$ with $(u_k)$ converges weakly to $u$ in $W^{1,p}(\Omega,\mathbb{R}^2)$. Then $\det(\nabla u_k)$ converges weakly to $\det(\nabla u)$ in $L^{p/2}(\Omega)$.

Remark: $\det(\nabla u_k)$ converges weakly to $\det(\nabla u)$ in $L^{p/2}(\Omega)$ means that $\lim\limits_{k\to\infty} \int_\Omega \det(\nabla u_k)vdx =\int_\Omega \det(\nabla u)vdx$ for all $v\in L^{\frac{p}{p-2}}(\Omega)$.

Proof of Reshetnyak:

1) At first one shows that $\lim\limits_{k\to\infty} \int_\Omega \det(\nabla u_k)vdx =\int_\Omega \det(\nabla u)vdx$ for all $v\in C_0^{\infty}(\Omega)$ (smooth functions with compact support in $\Omega$).

2) We use that $C_0^{\infty}(\Omega)$ is dense in $L^{\frac{p}{p-2}}(\Omega)$ with respect to $\|.\|_{L^q}$-norm, where $q=\frac{p}{p-2}$. This means, for all $\epsilon >0$, for all $v\in L^{q}(\Omega)$ there exists a $v_{\epsilon}\in C_0^{\infty}(\Omega)$ such that $\|v-v_{\epsilon}\|_{L^q}<\epsilon $. Claim: $\lim\limits_{k\to\infty} |\int_\Omega (\det(\nabla u_k)-\det(\nabla u))vdx|=0$.

Now I'm stuck with my understanding in the proof. To prove the claim in 2), at first they estimate $$|\int_\Omega \big(\det(\nabla u_k)-\det(\nabla u)\big)vdx|\le \|v-v_{\epsilon}\|_{L^q}\| \det(\nabla u_k)-\det(\nabla u)\|_{L^{p/2}}+|\int_\Omega \big(\det(\nabla u_k)-\det(\nabla u)\big)v_{\epsilon} dx|.$$

My question is: How do you get this estimation? Hölder's inequality is used somehow, but I don't get this... Thank you


1 Answer 1


Start from $$\int_\Omega \left(\det(\nabla u_k)-\det(\nabla u)\right)v\mathrm dx= \int_\Omega \left(\det(\nabla u_k)-\det(\nabla u)\right)\left(v-v_{\varepsilon}\right)\mathrm dx+\int_\Omega \left(\det(\nabla u_k)-\det(\nabla u)\right)v_{\varepsilon}\mathrm dx.$$ By the triangle inequality, $$\left|\int_\Omega \left(\det(\nabla u_k)-\det(\nabla u)\right)v\mathrm dx\right|\leqslant A+B,$$ where $A:=\left|\int_\Omega \left(\det(\nabla u_k)-\det(\nabla u)\right)\left(v-v_{\varepsilon}\right)\mathrm dx\right|$ and $B:=\left|\int_\Omega \left(\det(\nabla u_k)-\det(\nabla u)\right)v_{\varepsilon}\mathrm dx\right|$.

The term $A$ is controlled by Hölder's inequality applied with the exponents $p/2$ and $q$ and the functions $\det(\nabla u_k)-\det(\nabla u)$ and $v-v_{\varepsilon}$.


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