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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.

Date: Thursday, October 5, 2017
Time: 1:30 pm
Location: Wean Hall 7218
Submitted by:  Fonseca