##### Department of Mathematics,

University of California San Diego

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### Center for Computational Mathematics Seminar

## Ali Behzadan

#### UCSD

## On the Continuity of Exterior Differentiation Between Sobolev-Slobodeckij Spaces of Sections of Tensor Bundles on Compact Manifolds

##### Abstract:

Suppose $\Omega$ is a nonempty open set with Lipschitz continuous boundary in $\mathbb{R}^n$. There are certain exponents $e\in \mathbb{R}$ and $q\in (1, infty)$ for which $\displaystyle \frac{\partial}{\partial x^j}: W^{e,q}(\Omega) \rightarrow W^{e-1,q}(\Omega)$ is NOT a well-defined continuous operator. Now suppose $M$ is a compact smooth manifold. In this talk we will try to discuss the following questions: \begin{enumerate} \item How are Sobolev spaces of sections of vector bundles on $M$ defined? \item Is it possible to extend $d: C^\infty(M)\rightarrow C^\infty(T^{*}M)$ to a continuous linear map from $W^{e,q}(M)$ to $W^{e-1,q}(T^{*}M)$ for all $e\in \mathbb{R}$ and $q\in (1,\infty)$? \item Why are we interested in the above question? \end{enumerate}

### May 9, 2017

### 11:00 AM

### AP&M 2402

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