Siksha Sivaramakrishnan
Monash University
Biography
Siksha is a passionate mathematics student who has recently completed an education degree and is looking forward to starting honours in pure mathematics in 2019. She is interested in functional analysis and differential geometry and is excited to learn more about elliptic PDE theory.
Siksha is dedicated to communicating mathematics to the next generation and inciting a passion for mathematics in them with her education degree. She is also specifically interested in increasing the number of women in STEM fields.
Symmetrisations and Other Rearrangement Inequalities
Rearrangements improve the shape of a geometric object while preserving its size: a famous example is the Steiner symmetrisation, which gives an object a single axis of symmetry. These rearrangements can also preserve or improve analytic quantities, such as the energies of functions defined on these objects.
In this project we will examine a number of symmetric decreasing rearrangements such as Steiner symmetrisation, polarisation, Schwarz symmetrisation; and their behaviour via Hardy-Littlewood inequality, Riesz’s inequality, Polya-Szego inequality and so on. Applications include the Polya-Szego theorem, which says that the Dirichlet eigenvalue of a polygonal planar domain with no more than N sides is greater than the eigenvalue of the regular N-gon of the same area, for N = 3,4. (The case N ≥ 5 is an open question).
The motivation here is to support work on the N = 3 Polya-Szego conjecture under Robin boundary values. The conjecture is that the equilateral triangle minimises the first Robin eigenvalue among all triangles. This is an open problem. In this project, we aim is to construct explicit examples of Robin eigenfunctions on (non-equilateral) triangles and investigate whether symmetrisation does indeed decrease the eigenvalue, thus both supporting the conjecture, and supporting the use of symmetrisations as a useful technique in this context.
The difficulty in this approach arises from the non-zero boundary value: for general functions of this form the Polya-Szego inequality does not apply, but it may be possible to find useful monotonic quantities for the restricted class of Robin eigenfunctions.
Methodology:
• Review of key tools from measure theory, including Lp norms, and the co-area formula
• Study of recent literature on symmetric decreasing rearrangements
• Review the proof of the N = 3 Polya-Szego inequality with Dirichlet boundary conditions.
• Construct explicit examples of Robin eigenfunctions on triangles (either in closed form, or
numerically)
• Symmetrise the eigenfunctions, and calculate the Rayleigh quotient of the new function
References: Almut Burchard, A Short Course on Rearrangement Inequalities, 2009
Bernhard Kawohl, Rearrangements and Convexity of Level Sets in PDE, 1985
Evans and Gariepy, Measure Theory and Fine Properties of Functions, 1992.