##### Department of Mathematics,

University of California San Diego

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### Math 248 - Real Analysis

## Igor Kukavica

#### USC

## On the Size of the Nodal Sets of Solutions of Elliptic and Parabolic PDEs

##### Abstract:

We present several results on the size of the nodal (zero) set for solutions of partial differential equations of elliptic and parabolic type. In particular, we show a sharp upper bound for the $(n-1)$-dimensional Hausdorff measure of the nodal sets of the eigenfunctions of regular analytic elliptic problems in ${\mathbb R}^n$. We also show certain more recent results concerning semilinear equations (e.g. Navier-Stokes equations) and equations with non-analytic coefficients. The results on the size of nodal sets are connected to quantitative unique continuation, i.e., on the estimate of the order of vanishing of solutions of PDEs at a point. The results on unique continuation are joint with Ignatova and Camliyurt.

Andrej Zlatos

### November 21, 2017

### 8:45 AM

### AP&M 7321

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