PDE Seminar Abstracts

Barbara Brandolini

Università degli Studi di Napoli “Federico II”, Italy

Mon 19th Aug 2019, 12-1pm, Carslaw Room 829 (AGR)

Università degli Studi di Napoli “Federico II”, Italy

Mon 19th Aug 2019, 12-1pm, Carslaw Room 829 (AGR)

Let $\Omega \subset {\mathbb{R}}^{n}$ be a convex domain and let $f:\Omega \to \mathbb{R}$ be a positive, subharmonic function (i.e. $\Delta f\ge 0$). Then

$$\frac{1}{\left|\Omega \right|}{\int}_{\Omega}f\phantom{\rule{0.3em}{0ex}}dx\le \frac{{c}_{n}}{\left|\partial \Omega \right|}{\int}_{\partial \Omega}f\phantom{\rule{0.3em}{0ex}}d\sigma ,$$

where ${c}_{n}\le 2{n}^{3\u22152}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies ${c}_{n}\ge n-1$. As a byproduct, we establish the following sharp geometric inequality for two convex domains where one contains the other ${\Omega}_{2}\subset {\Omega}_{1}\subset {\mathbb{R}}^{n}$:

$$\frac{\left|\partial {\Omega}_{1}\right|}{\left|{\Omega}_{1}\right|}\frac{\left|{\Omega}_{2}\right|}{\left|\partial {\Omega}_{2}\right|}\le n.$$

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