H. Render

will speak on

Fischer decomposition for entire functions and the Dirichlet problem for unbounded quadratic surfaces

Time: 3:00PM
Date: Tue 19th April 2022
Location: Seminar Room SCN 1.25 [map]

Further information

Abstract: Let $P_{2k}$ be a homogeneous non-negative polynomial of degree $2k$ and assume that $P_j$ for $j=0,...,\beta <2k$ are homogeneous polynomials of degree $j$. Further a certain integral inequality depending on a parameter \alpha and $P_2k$ is assumed which is valid for all homogeneous polynomials of degree $m$.

The main result of the talk states that for any entire function f of order $\rho < ( 2k-\beta ) / \alpha$ there exist entire functions q and r of finite order with
$$
f= ( P_2k-P_\beta -....-P_0 ) q+r
$$
and $\Delta ^{k}r=0$
where $\Delta$ is the Laplace operator.


This result is used to establish the existence of entire harmonic solutions of the Dirichlet problem for cylinders and parabola-shaped domains for data given by entire functions of order smaller than $1$ and $1/2$ respectively.

https://ucd-ie.zoom.us/j/64933463319

(This talk is part of the Analysis series.)

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