# Analysis and Partial Differential Equations

## Joint Seminar Day

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.

## Venue:

University of Wollongong: See the information on how to get there. All Talks are in Building 3 Room 224

## Program

Time Speaker Morning session Ben Andrews (ANU) Horospherically convex hypersurfaces, geometric inequalities, and conformal deformation Phung Trong Thuc (UOW) The Bergman kernel and related problems Qi-Rui Li (ANU) A class of Monge–Ampère equations on the sphere Adam Sikora (Macquarie) Riesz transforms on a class of non-doubling manifolds Afternoon session Quôc Thông Lê Gia (UNSW) A non-uniform discretization of stochastic heat equations with multiplicative noise on the unit sphere Florica Cîrstea (USyd) Gradient estimates for nonlinear elliptic equations with a gradient-dependent nonlinearity Debopriya Mukherjee (UNSW) Study of stochastic viscoelastic fluid models and related constrained physical problems Frederic Weber (Ulm, Germany) The $\Gamma$-Calculus of Bakry-Emery and $\Phi$-Entropy Inequalities Minh Dao (Newcastle) On regularisation with Bregman–Moreau envelopes

## Abstracts of Talks

### Horospherically convex hypersurfaces, geometric inequalities, and conformal deformation

Ben Andrews (Australian National University)

#### Abstract

I will discuss the proof of some new geometric inequalities for horospherically convex hypersurfaces in hyperbolic space, proved using geometric flows. Along the way I will discuss some of the interesting structure of hyperbolic hypersurfaces, including horospherical support functions and horospherical Gauss maps. Finally, I will mention a curious connection between horospherical hypersurfaces and conformally flat metrics on the sphere, which gives a direct correspondence linking flows of hypersurfaces by curvature functions to conformal deformation of conformally flat metrics by functions of the Schouten tensor.

Joint work with Xuzhong Chen (Hunan University) and Yong Wei (ANU)

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Florica Cîrstea (University of Sydney)

#### Abstract

In this talk, we will present gradient estimates of the positive solutions to weighted $p$-Laplacian type equations with a gradient-dependent nonlinearity of the form

 (1)

Here, $\Omega \subseteq {ℝ}^{N}$ denotes a domain containing the origin with $N\ge 2$, whereas $m,q\in \left[0,\infty \right)$, $1 and $q>max\left\{p-m-1,\sigma +\tau -1\right\}$. The main difficulty arises from the dependence of the right-hand side of (1) on $x$, $u$ and $|\nabla u|$, without any upper bound restriction on the power $m$ of $|\nabla u|$. Our proof of the gradient estimates is based on a two-step process relying on a modified version of the Bernstein’s method. As a by-product, we extend the range of applicability of the Liouville-type results known for (1).

This is joint work with Joshua Ching (University of Sydney).

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### On regularisation with Bregman–Moreau envelopes

Minh N. Dao (University of Newcastle)

#### Abstract

Fifty years ago, Attouch proposed to use the Moreau envelope for regularisation. Since then, this branch of convex analysis has developed in many fruitful directions. In a different development, in 1967, Bregman introduced what is nowadays the Bregman distance as a measure of discrepancy between two points generalising the square of the Euclidean distance. Proximity operators based on the Bregman distance have become a topic of significant research as they are useful in algorithmic solution of optimisation problems. In this work, we complement these two different strands of convex analysis by systematically investigating regularisation aspects of the Bregman–Moreau envelope. We also present various natural extensions of classical results and illustrate them with examples.

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### A non-uniform discretization of stochastic heat equations with multiplicative noise on the unit sphere

Quôc Thông Lê Gia (University of New South Wales)

#### Abstract

In this work, we investigate the discretization of a class of stochastic heat equations on the unit sphere with multiplicative noises. A spectral method is used for spatial discretization while an implicit Euler scheme with non-uniform timestep is used for time discretization. Some numerical experiments inspired by Earth’s surface temperature data analysis GISTEMP provided by NASA will be given.

This is a joint work with Yoshihito Kazashi (UNSW)

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### Study of stochastic viscoelastic fluid models and related constrained physical problems

Debopriya Mukherjee (University of New South Wales)

#### Abstract

I will discuss about the existence of a solution to an optimal relaxed control problem for the linearly-coupled viscoelastic Oldroyd-B model driven by Levy noise. Then, the existence and uniqueness of local (maximal) strong solutions for the nonlinearly-coupled critical stochastic viscoelastic model in both two and three dimensions will be discussed. Furthermore, I will discuss about the weak (martingale) solutions to the nonlinearly-coupled critical and sub-critical stochastic Oldroyd-B model. In addition, I will prove the existence of an invariant measure for the sub-critical problem, using bw-Feller property of the associated Markov semigroup in a Poincare domain in two-dimensions.

Finally, I will move to the study on the effects of magnetization dynamics inside a ferromagnetic materials at low temperature taking values in a three-dimensional sphere in the form of the Landau-Lifshitz-Gilbert equations and prove the existence of a strong solution of the one-dimensional stochastic problem via Wong-Zakai approximation.

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### A class of Monge–Ampère equations on the sphere

Qi-Rui Li (Australian National University)

#### Abstract

There are a number of geometric problems which can be reduced to the study of the Monge–Ampere equation on the sphere, including the Aleksandrov problem, the Minkowski problem, and more generally the ${L}_{p}$ dual Minkowski problem introduced mostly recently by Lutwak-Yang-Zhang. In this talk we give a brief discuss on these problems.

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### Riesz transforms on a class of non-doubling manifolds

#### Abstract

We consider a certain class of manifolds $M$ which do not satisfies the doubling condition. We completely describe the range of exponents $p$ for which the Riesz transform on $M$ is a bounded operator on ${L}^{p}\left(M\right)$.

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### The Bergman kernel and related problems

Phung Trong Thuc (Unversity of Wollongong)

#### Abstract

In this talk, ${L}^{2}$ boundary estimates for the Bergman kernel and the ${L}^{p}$ regularity problem for Bergman-Toeplitz operators are discussed.

This talk is joint work with Jiakun Liu and Tran Vu Khanh.

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### The $\Gamma$-Calculus of Bakry–Emery and $\Phi$-Entropy Inequalities

Frederic Weber (Universität Ulm, Germany)

#### Abstract

In this talk we investigate the so called $\Gamma$-Calculus for Markov Diffusion operators, which we will motivate by the modell example of elliptic second order differential operators. We will discuss it’s application to obtain a family of functional inequalities, the so called $\Phi$-Entropy inequalities, among which are for instance Poincare Inequalities and Logarithmic Sobolev inequalities. We conclude this talk with applications to the long time behaviour of Fokker-Planck equations.

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## Organisers

• Ben Andrews (ANU)
• Daniel Daners (USyd)
• Xuan Duong (Macquarie)
• Ian Doust (UNSW)
• Daniel Hauer (USyd)
• Ji Li (Macquarie)