Inequality involving BV norm and a regularizing kernel - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T19:41:17Z http://mathoverflow.net/feeds/question/90029 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90029/inequality-involving-bv-norm-and-a-regularizing-kernel Inequality involving BV norm and a regularizing kernel Beni Bogosel 2012-03-02T11:51:32Z 2012-03-02T11:51:32Z <p>In the same article by Benoit Perthame: <a href="http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/#" rel="nofollow">http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/#</a> (related to this question <a href="http://mathoverflow.net/questions/89801/a-limit-involving-a-regularizing-kernel" rel="nofollow">http://mathoverflow.net/questions/89801/a-limit-involving-a-regularizing-kernel</a>) I encountered an inequality, which I didn't manage to prove.</p> <p>It is like this:</p> <p>$$\int_{\Bbb{R}^d}\left[ |u^0(x)|-\int_{\Bbb{R}}\left(\chi(\xi,u^0(x))\star \varphi_\varepsilon \right)^2d \xi\right]dx \leq C \|u^0\|_{BV}\cdot\varepsilon$$</p> <p>where $$\chi(\xi,u)=\begin{cases} 1 &amp; {0\leq \xi\leq u} \newline -1 &amp; u \leq \xi \leq 0 \newline 0 &amp; \text{otherwise} \end{cases}$$</p> <p>and $\varphi_\varepsilon$ is a regularization kernel in $x$ and $u_0$ is regular enough for all objects to be well defined.</p> <p>In the article, the inequality is stated as obvious, and no indication, reference or attempt to prove it is made. It is possible to prove that the LHS tends to zero as $\varepsilon \to 0$. Still, I cannot get the majorization by $\varepsilon$ in the RHS, which clearly depends only on $\varphi_{\varepsilon}$.</p> <p>Do you have some ideas in proving this inequality? Thank you.</p>