Discrete functional inequalities on lattice graphs
File(s)
Author(s)
Gupta, Shubham
Type
Thesis
Abstract
In this thesis, we study problems at the interface of analysis and discrete mathematics. We discuss analogues of well known Hardy-type inequalities and Rearrangement inequalities on the lattice graphs Z^d, with a particular focus on behaviour of sharp constants and optimizers.
In the first half of the thesis, we analyse Hardy inequalities on Z^d, first for d=1 and then for d >= 3. We prove a sharp weighted Hardy inequality on integers with power weights of the form n^\alpha. This is done via two different methods, namely 'super-solution' and 'Fourier method'. We also use Fourier method to prove a weighted Hardy type inequality for higher order operators. After discussing the one dimensional case, we study the Hardy inequality in higher dimensions (d >= 3). In particular, we compute the asymptotic behaviour of the sharp constant in the discrete Hardy inequality, as d \rightarrow \infty. This is done by converting the inequality into a continuous Hardy-type inequality on a torus for functions having zero average. These continuous inequalities are new and interesting in themselves.
In the second half, we focus our attention on analogues of Rearrangement inequalities on lattice graphs. We begin by analysing the situation in dimension one. We define various notions of rearrangements and prove the corresponding Polya-Szego inequality. These inequalities are also applied to prove some weighted Hardy inequalities on integers. Finally, we study Rearrangement inequalities (Polya-Szego) on general graphs, with a particular focus on lattice graphs Z^d, for d >=2. We develop a framework to study these inequalities, using which we derive concrete results in dimension two. In particular, these results develop connections between Polya-Szego inequality and various isoperimetric inequalities on graphs.
In the first half of the thesis, we analyse Hardy inequalities on Z^d, first for d=1 and then for d >= 3. We prove a sharp weighted Hardy inequality on integers with power weights of the form n^\alpha. This is done via two different methods, namely 'super-solution' and 'Fourier method'. We also use Fourier method to prove a weighted Hardy type inequality for higher order operators. After discussing the one dimensional case, we study the Hardy inequality in higher dimensions (d >= 3). In particular, we compute the asymptotic behaviour of the sharp constant in the discrete Hardy inequality, as d \rightarrow \infty. This is done by converting the inequality into a continuous Hardy-type inequality on a torus for functions having zero average. These continuous inequalities are new and interesting in themselves.
In the second half, we focus our attention on analogues of Rearrangement inequalities on lattice graphs. We begin by analysing the situation in dimension one. We define various notions of rearrangements and prove the corresponding Polya-Szego inequality. These inequalities are also applied to prove some weighted Hardy inequalities on integers. Finally, we study Rearrangement inequalities (Polya-Szego) on general graphs, with a particular focus on lattice graphs Z^d, for d >=2. We develop a framework to study these inequalities, using which we derive concrete results in dimension two. In particular, these results develop connections between Polya-Szego inequality and various isoperimetric inequalities on graphs.
Version
Open Access
Date Issued
2023-06
Date Awarded
2023-08
Copyright Statement
Creative Commons Attribution NonCommercial NoDerivatives Licence
Advisor
Laptev, Ari
Sponsor
President's PhD scholarship, Imperial College London
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)