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Related Experiment Video

Updated: Feb 8, 2026

Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations
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Maximum principles and Bôcher type theorems.

Congming Li1,2, Zhigang Wu3, Hao Xu4,2

  • 1School of Mathematical Sciences and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China; congming.li@sjtu.edu.cn.

Proceedings of the National Academy of Sciences of the United States of America
|June 22, 2018
PubMed
Summary

This study establishes maximum principles and Bôcher-type theorems for nonnegative superharmonic and fractional superharmonic functions. A key finding is the connection between maximum principles and Bôcher-type theorems for these functions.

Keywords:
Bôcher theoremfractional Laplaciansingular solution

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Area of Science:

  • Analysis and Partial Differential Equations
  • Potential Theory

Background:

  • Superharmonic functions are fundamental in potential theory.
  • Fractional calculus extends classical analysis, introducing new function classes like fractional superharmonic functions.
  • Understanding function behavior on specific domains, such as punctured balls, is essential for theoretical advancements.

Purpose of the Study:

  • To establish maximum principles for nonnegative superharmonic and fractional superharmonic functions on a punctured ball.
  • To establish Bôcher-type theorems for these function classes on the same domain.
  • To explore and highlight the relationship between maximum principles and Bôcher-type theorems.

Main Methods:

  • Development of analytical techniques tailored for superharmonic and fractional superharmonic functions.
  • Application of techniques to analyze function behavior near singularities within the punctured ball domain.
  • Formal derivation of inequalities and limit behaviors characteristic of maximum principles and Bôcher-type theorems.

Main Results:

  • Establishment of novel maximum principles for nonnegative superharmonic functions on a punctured ball.
  • Establishment of analogous maximum principles for fractional superharmonic functions.
  • Proof of Bôcher-type theorems for both superharmonic and fractional superharmonic functions.
  • Demonstration of a significant link between maximum principles and Bôcher-type theorems.

Conclusions:

  • The established maximum principles provide essential bounds and properties for superharmonic and fractional superharmonic functions.
  • Bôcher-type theorems offer insights into the singularity behavior of these functions.
  • The crucial observation is the inherent connection between maximum principles and Bôcher-type theorems, unifying these concepts in the context of fractional calculus and potential theory.