# Analysis and Partial Differential Equations

The aim of this seminar day is to bring together twice a year specialists, early career researchers and PhD students working in analysis, partial differential equations and related fields in Australia, in order to report on research, fostering contacts and to begin new research projects between the participants.

This seminar day is organised jointly with the related research groups of the Australian National University, Macquarie University, University of Sydney, University of Wollongong, UNSW and University of Newcastle.

In particular, this event has the intention to give PhD students and early career researchers the opportunity to present their research to a wider audience.

If you are interested in attending please register

## Venue:

Australian National University, Canberra. All talks are in the John Dedman Building in Room G35

The program is designed so that participants from outside the ACT can travel in the morning of 9 February to Canberra, and then back in the afternoon of 10 February. Here is a list of places that are within reasonable walking distance from ANU: Hotel-Hotel, Medina Serviced Apartments Canberra, James Court, Capital Executive Apartment Hotel, University House. More options at agoda.com, expedia.com, bookings.com, Visit Canberra Explore 141.

Here is some information on how to get to Canberra.

## Tentative Program

Tuesday 9 February

14:30–15:35 - Welcome
14:35–15:20 - Mark Veraar (Guest Speaker, TU Delft, visiting ANU)
A new approach to maximal regularity for parabolic PDEs
15:25–16:00 - Afternoon Tea
16:00–16:45 - Anna Tomskova (University of New South Wales)
Fréchet differentiability of the norm of non-commutative $$L_p$$-spaces
16:50–17:25 - Piotr Rybka (University of Warsaw, Poland, visiting UOW)
Special cases of the planar least gradient problem
19:00–21:00 - Dinner at the pizzeria Debacle

Wednesday 10 February

09:00–09:45 - Lixin Yan (Guest Speaker, Sun Yat Sen University, China, visiting Macquarie)
Multicommutators and Multiplier Theorems
09:50–10:20 - Morning Tea
10:20–11:05 - Yanqin Fang (Wollongong)
Liouville theorems involving the fractional Laplacian
11:10–11:55 - Andrew Hassell (Australian National University)
Upper and lower bounds on boundary values of Neumann eigenfunctions, and applications

You can also download the schedule (PDF)

## Abstracts of Talks

### Liouville theorems involving the fractional Laplacian

Yanqin Fang (University of Wollongong)

#### Abstract

The fractional Laplacian in Euclidean space is a non-local operator. We establish Liouville type theorems, non-existence of positive solutions of Dirichlet problem involving the fractional Laplacian. We obtain the equivalence between a partial differential equation and an integral equation from the uniqueness of harmonic functions on a half space. Applying the method of moving planes in integral forms, we prove the non-existence of positive solutions of an integral equation.

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### Upper and lower bounds on boundary values of Neumann eigenfunctions, and applications

Andrew Hassell (Australian National University)

#### Abstract

For smooth bounded Euclidean domains, we prove upper and lower ${L}^{2}$ bounds on the boundary data of Neumann eigenfunctions, and prove quasi-orthogonality of this boundary data in a spectral window. The bounds are tight in the sense that both are independent of eigenvalue; this is achieved by working with an appropriate norm for boundary functions, which includes a ‘spectral weight’, that is, a function of the boundary Laplacian. This spectral weight is chosen to cancel concentration at the boundary that can happen for ‘whispering gallery’ type eigenfunctions.

As an application, we give bounds on the distance from an arbitrary positive real number $E>0$ to the nearest Neumann eigenvalue, in terms of boundary normal-derivative data of a trial function $u$ solving the Helmholtz equation $\left(\Delta -E\right)u=0$. These improve over previously known bounds by a factor of ${E}^{5∕6}$. This can be used to design an improved “method of particular solutions” for finding Neumann eigenfunctions numerically.

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### Special cases of the planar least gradient problem

Piotr Rybka (University of Warsaw, Poland)

#### Abstract

We study the least gradient problem in two special cases:

1. the natural boundary conditions are imposed on a part of the stricly convex domain;
2. the Dirichlet type data are imposed on a boundary of a rectangle.

We show the existence of solutions and study properties of solutions for special cases of the data.

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### Fréchet differentiability of the norm of non-commutative ${L}_{p}$-spaces

Anna Tomskova (University of New South Wales)

#### Abstract

Let $M$ be a von Neumann algebra and let $\left({L}_{p}\left(M\right),\parallel .{\parallel }_{p}\right)$, $1\le p<\infty$ be Haagerup’s ${L}_{p}$-space on $M$. The main ideas of the proof that the differentiability properties of $\parallel .{\parallel }_{p}$ are precisely the same as those of classical (commutative) ${L}_{p}$-spaces are presented. Our main instruments are the theories of multiple operator integrals and singular traces.

This is joint work with Fedor Sukochev, Denis Potapov and Dimitry Zanin.

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### A new approach to maximal regularity for parabolic PDEs

Mark Veraar (TU Delft, Netherlands)

#### Abstract

Maximal regularity can often be used to obtain a priori estimates which give global existence results. In this talk I will explain a new approach to maximal ${L}^{p}$-regularity for parabolic PDEs with time dependent generator $A\left(t\right)$. Here we do not assume any continuity properties of $A\left(t\right)$ as a function of time. We show that there is an abstract operator theoretic condition on A(t) which is sufficient to obtain maximal ${L}^{p}$-regularity. As an application I will obtain an optimal ${L}^{p}\left({L}^{q}\right)$ regularity result in the case each $A\left(t\right)$ is a system of $2m$-th order elliptic differential operator on ${ℝ}^{d}$ in non-divergence form. The main novelty is that the coefficients are merely measurable in time and we allow the full range $1. This talk is based on joint work with Chiara Gallarati.

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### Multicommutators and Multiplier Theorems

Lixin Yan (Sun Yat-Sen University, China)

#### Abstract

We obtain the boundedness of the $n$-th dimensional Calderón-Coifman-Journé multicommutator from ${L}^{{p}_{0}}\left({ℝ}^{n}\right)×{L}^{{p}_{1}}\left({ℝ}^{n}\right)×\cdots ×{L}^{{p}_{k}}\left({ℝ}^{n}\right)$ to ${L}^{p}\left({ℝ}^{n}\right)$ in the largest possible open set of indices $\left(\frac{1}{{p}_{0}},\frac{1}{{p}_{1}},\dots ,\frac{1}{{p}_{k}}\right)$ with $\frac{1}{{p}_{0}}+\frac{1}{{p}_{1}}+\cdots +\frac{1}{{p}_{k}}=\frac{1}{p}$, which is the range $\frac{1}{k+1}. The proof exploits the limited smoothness of the symbol of the multicommutator via a new multilinear multiplier theorem for symbols of restricted smoothness which lie locally in certain Sobolev spaces. Our multiplier approach to this problem is a new contribution in the understanding of Calderón’s commutator program.

This is a joint work with Loukas Grafakos, Danqing He and Hanh Van Nguyen.

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## Organisers

 Daniel Daners (USyd) Xuan Duong (Macquarie) Daniel Hauer (USyd) Jeff Hogan (Newcastle) Ji Li (Macquarie) James McCoy (Wollongong) Pierre Portal (ANU) Adam Sikora (Macquarie) Glen Wheeler (Wollongong) Valentina-Mira Wheeler (Wollongong)