##### Department of Mathematics,

University of California San Diego

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### Math 248 - Analysis Seminar

## Jonas Hirsch

#### University of Leipzig

## On bounded solutions of linear elliptic operators with measurable coefficients - De Giorgiâ€™s theorem revisited

##### Abstract:

We consider the classical framework of the famous De-Giorgi-Nash-Moser theorem: $div(A(x)\nabla u)=f$, where $A(x)$ is a symmetric, elliptic matrix field, $f$ is given and $u:U\subset \mathbb{R}^n\to\mathbb{R}$ is the unknown. N. Trudinger was the first one to relax the assumptions on the coefficients matrix $A(x)$. He was able to derive boundedness results if the matrix is barely integrable in the right spaces. In particular he was able to show that if $\lambda(x)|\xi|^2\leq \xi\cdot A(x)\xi\leq \Lambda(x)|\xi|^2,\quad \forall x$ and the $\lambda^{-1}\in L^p, \Lambda\in L^q$ satisfying $\frac{1}{p}+\frac{1}{q}<\frac{2}{n}$. The integrability condition had been considerably improved by P. Bella and M. Schaffner in the framework of the Moser-iteration to $\frac{1}{p}+\frac{1}{q}<\frac{2}{n-1}$. A counterexample had been constructed by Franchi, Serapioni, and Serra Cassano under $\frac{1}{p}+\frac{1}{q}>\frac{2}{n}$. The aim of this talk is to revisit De Giorgiâ€™s original approach having in mind the question concerning the optimal integrability assumption on the coefficient field. We will present how this question is surprisingly linked to a question in linear programming with an infinite horizon. This talk will be about my ongoing project with M. Schaffner, hence about work in progress.

### October 26, 2021

### 11:00 AM

https://ucsd.zoom.us/j/99515535778

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