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Calculus of variations on hypergraphs

Mengqiu Shao, Yulu Tian, Liang Zhao

Abstract

We have established a coherent framework for applying variational methods to partial differential equations on hypergraphs, which includes the propositions of calculus and function spaces on hypergraphs. Several results related to the maximum principle on hypergraphs have also been proven. As applications, we demonstrated how these can be used to study partial differential equations on hypergraphs.

Calculus of variations on hypergraphs

Abstract

We have established a coherent framework for applying variational methods to partial differential equations on hypergraphs, which includes the propositions of calculus and function spaces on hypergraphs. Several results related to the maximum principle on hypergraphs have also been proven. As applications, we demonstrated how these can be used to study partial differential equations on hypergraphs.
Paper Structure (7 sections, 18 theorems, 109 equations, 1 figure)

This paper contains 7 sections, 18 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Calculus on hypergraphs
  3. Maximum Principles on hypergraphs
  4. Partial differential equations on hypergraphs
  5. Linear Schrödinger equation
  6. Yamabe type equation
  7. Nonlinear Schrödinger equation

Key Result

Lemma 2.1

For any $\phi, \psi\in L^2(V)$, there holds In particular, we have where $\|\nabla\phi\|_{\vec{E}}^2:=\langle\nabla \phi,\nabla \phi\rangle_{\vec{E}}$.

Figures (1)

  • Figure 1: The hypergraph $H_{4}$

Theorems & Definitions (39)

  • Example 2.1
  • Remark 2.2
  • Definition 2.1
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 29 more