##### Department of Mathematics,

University of California San Diego

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### Math 248 - Real Analysis

## Matthew Blair

#### University of New Mexico

## $L^{p}$ Bounds on Restrictions of Eigenfunctions in Rough Media

##### Abstract:

Given a compact Riemannian manifold, it is well known that the Laplace-Beltrami operator has a sequence of eigenfunctions which form an orthonormal basis for the square integrable functions. A problem of recent interest has been to prove $L^{p}$ estimates on these eigenfunctions, which bound their $L^{p}$ norm by a power of the corresponding eigenvalue, illuminating their size and concentration properties. We will investigate the validity of these bounds for manifolds equipped with rough metrics, such as those of Lipschitz regularity, which is significant in extending the theory to domains in $R^{n}$ and manifolds with boundary. In particular, we discuss a recent result regarding $L^{n}$ estimates on the restriction of these eigenfunctions to submanifolds.

Host: Jacob Sterbenz

### January 23, 2013

### 12:00 PM

### AP&M 5829

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