Discrete Functional Inequalities on Lattice Graphs Explored for PhD Thesis

Concetti Chiave

Studying discrete analogues of Hardy-type inequalities on lattice graphs reveals sharp constants and optimizers behavior.

Sintesi

The thesis delves into discrete functional inequalities on lattice graphs, focusing on Hardy-type and Rearrangement inequalities. The study explores the behavior of sharp constants and optimizers in one-dimensional and higher-dimensional cases. Methods like super-solution and Fourier transform are employed to prove weighted Hardy inequalities for specific values of α and β. The results reveal insights into the fundamental differences between continuous and discrete Hardy inequalities in higher dimensions.

Statistiche

(2.1) contains a two-parameter family of weighted Hardy inequalities.
(2.7) presents a special case with power weights for specific α values.
(2.8) offers an improvement for α ∈ [1/3, 1) ∪ {0}.

Citazioni

Approfondimenti chiave tratti da

Discrete functional inequalities on lattice graphs

by Shubham Gupt... alle arxiv.org 03-18-2024

https://arxiv.org/pdf/2403.10270.pdf

Domande più approfondite

What implications do the findings have for spectral theory applications

The findings in the research on discrete Hardy inequalities have significant implications for spectral theory applications. In particular, these inequalities play a crucial role in studying the localization properties of eigenfunctions of discrete Schrödinger operators. By understanding and establishing sharp constants in weighted Hardy-type inequalities on lattice graphs, researchers can gain insights into the decay properties of eigenfunctions associated with these operators. This knowledge is essential for analyzing various physical phenomena where spectral properties are involved.

How do the limitations of calculus in the discrete setting impact the study of higher-dimensional Hardy inequalities

The limitations of calculus in the discrete setting pose challenges when studying higher-dimensional Hardy inequalities. Unlike in continuous spaces where calculus tools are readily available, transferring techniques from the continuum to discrete settings becomes more complex due to this limitation. The absence of polar coordinates and other fundamental differences between continuous and discrete settings make it harder to extend results from one dimension to higher dimensions. These limitations require innovative approaches and methodologies tailored specifically for discrete spaces.

How can the insights gained from this research be applied to other areas beyond mathematics

The insights gained from this research on discrete functional inequalities can be applied beyond mathematics to various fields such as computer science, physics, engineering, and data analysis. For example:

Computer Science: Understanding optimization problems on graphs using rearrangement techniques can improve algorithms related to network flow optimization or graph partitioning.
Physics: Applying weighted Hardy-type inequalities can enhance studies involving quantum mechanics or statistical mechanics.
Engineering: Utilizing concepts from spectral theory applications can aid in signal processing or image recognition algorithms.
Data Analysis: Techniques developed for analyzing functions on lattice graphs can be valuable for pattern recognition tasks or anomaly detection in large datasets.
These interdisciplinary applications demonstrate how mathematical research on functional inequalities transcends traditional boundaries and contributes to diverse areas of study.