Center for                           Nonlinear Analysis CNA Home People Seminars Publications Workshops and Conferences CNA Working Groups CNA Comments Form Summer Schools Summer Undergraduate Institute PIRE Cooperation Graduate Topics Courses SIAM Chapter Seminar Positions Contact CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium Monica Torres Purdue University Title: Characterizations of measures in the dual of $BV$ and related isometric isomorphisms Abstract: The solvability of the equation div ${\boldsymbol F} = \mu$ is connected to the analysis of $BV^*$, the dual of the space of functions of bounded variation. It is an open problem in geometric measure theory to characterize all the elements of $BV^*$. In this talk we present the characterization of all (signed) measures in $BV_{\frac{n}{n-1}}({\mathbb{R}}^n)^*$, where $BV_{\frac{n}{n-1}}({\mathbb{R}}^n)$ is defined as the space of all functions $u$ in $L^{\frac{n}{n-1}}({\mathbb{R}}^n)$ such that $Du$ is a finite vector-valued measure. Moreover, we show that the measures in $BV_{\frac{n}{n-1}}({\mathbb{R}}^n)^*$ coincide with the measures in $\dot W^{1,1}({\mathbb{R}}^n)^*$. %the dual of the homogeneous Sobolev space $\dot W^{1,1}({\mathbb{R}}^n)$, in the sense of isometric isomorphism. The space $\dot W^{1,1}({\mathbb{R}}^n)^*$ is known as the $G$ space in image processing and it plays a key role in modeling the noise of an image. Our results also provide a formula for $\left \Vert \mu \right \Vert_{G}$.As a consequence of our characterizations, an old issue raised by Meyers and Ziemer is resolved by constructing a locally integrable function $f$ such that $f$ belongs to $BV({\mathbb{R}}^n)^{*}$ but $|f|$ does not belong to $BV({\mathbb{R}}^n)^{*}$ (we will show that $BV({\mathbb{R}}^n)^{*}$ and $BV_{\frac{n}{n-1}}({\mathbb{R}}^n)^*$ are isometrically isomorphic). For a bounded open set $\Omega$ with Lipschitz boundary, we characterize the measures in the dual space $BV_0(\Omega)^*$ where $BV_0(\Omega)$ is the space of functions of bounded variation with zero trace on the boundary of $\Omega$. We show that the measures in $BV_0(\Omega)^*$ coincide with the measures in $W^{1,1}_0(\Omega)^*$.Joint work with Nguyen Cong Phuc.Recording: http://mm.math.cmu.edu/recordings/cna/MonicaTorres_small.mp4Date: Thursday, October 5, 2017Time: 1:30 pmLocation: Wean Hall 7218Submitted by:  Fonseca