Geometric variational problems and nonlinear partial differential systems. We will investigate several important problems on non-linear partial differential systems, bridging analysis, differential geometry and mathematical physics. Harmonic maps are the prototype of maps minimizing the Dirichlet energy. The liquid crystal configuration generalizes the harmonic map with values into two dimensional spheres. The Yang-Mills equations originated from particle physics. We will make fundamental contri ....Geometric variational problems and nonlinear partial differential systems. We will investigate several important problems on non-linear partial differential systems, bridging analysis, differential geometry and mathematical physics. Harmonic maps are the prototype of maps minimizing the Dirichlet energy. The liquid crystal configuration generalizes the harmonic map with values into two dimensional spheres. The Yang-Mills equations originated from particle physics. We will make fundamental contributions to these topics: Regularity problem and energy minimality of weakly harmonic maps, Weak solutions of the liquid crystal equilibrium system, Yang-Mills heat flow and singular Yang-Mills connections.
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Singular phenomena for nonlinear partial differential equations arising in applications. The development of nonlinear Partial Differential Equations (PDEs) in Australia is recognized worldwide through the outstanding contributions of mathematicians from the ANU, University of Sydney and other top Australian Universities. This project undertakes research in the PDEs field and follows directions of very current interest at an international level. Beyond the ANU, the project will enhance expertise ....Singular phenomena for nonlinear partial differential equations arising in applications. The development of nonlinear Partial Differential Equations (PDEs) in Australia is recognized worldwide through the outstanding contributions of mathematicians from the ANU, University of Sydney and other top Australian Universities. This project undertakes research in the PDEs field and follows directions of very current interest at an international level. Beyond the ANU, the project will enhance expertise in Australia in very active areas of mathematics research related to applications in physics, biology and other applied disciplines. Moreover, it will foster collaboration with mathematicians of international standing from Australia and abroad. Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE140101366
Funder
Australian Research Council
Funding Amount
$376,527.00
Summary
Fully nonlinear partial differential equations in optimisation and applications. Fully nonlinear partial differential equations of Monge-Ampere type and their applications in optimal transportation have been studied intensively in the past two decades. Optimal transportation is a subject in linear optimisation. This project will develop a new theory on Monge-Ampere type equations in nonlinear optimisation, which is a much broader area with many significant applications. This project will investi ....Fully nonlinear partial differential equations in optimisation and applications. Fully nonlinear partial differential equations of Monge-Ampere type and their applications in optimal transportation have been studied intensively in the past two decades. Optimal transportation is a subject in linear optimisation. This project will develop a new theory on Monge-Ampere type equations in nonlinear optimisation, which is a much broader area with many significant applications. This project will investigate fundamental properties of solutions to this problem and explore further real-world applications. The outcomes of this project will have a substantial impact on partial differential equations and related research subjects. Read moreRead less
Monge-Ampere equations and applications. The Monge-Ampere equation is a premier fully nonlinear partial differential equation with significant applications in geometry, physics and applied science. Building upon breakthroughs made by the proposers in previous grant research, this project aims to resolve challenging problems involving Monge-Ampere type equations and applications. The project goal is to establish new regularity theory and classify singularity profile for solutions to Monge-Ampere ....Monge-Ampere equations and applications. The Monge-Ampere equation is a premier fully nonlinear partial differential equation with significant applications in geometry, physics and applied science. Building upon breakthroughs made by the proposers in previous grant research, this project aims to resolve challenging problems involving Monge-Ampere type equations and applications. The project goal is to establish new regularity theory and classify singularity profile for solutions to Monge-Ampere type equation arising in applied sciences, by introducing new ideas and developing innovative cutting-edge techniques. Expected outcomes include resolution of outstanding open problems and continuing enhancement of Australian leadership and expertise in a major area of mathematics.
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Analysis of nonlinear partial differential equations describing singular phenomena. This project will advance knowledge on a huge variety of systems with applications across the sciences by providing new methods to investigate nonlinear partial differential equations with singularities. The analysis of many models describing physical and biological systems relies on such equations.