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Monotonicity formulae for variational problems.

Lawrence C Evans1

  • 1Department of Mathematics, University of California, , Berkeley, CA, USA.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|November 20, 2013
PubMed
Summary
This summary is machine-generated.

This paper explores monotonicity formulas for variational problems. These formulas offer useful insights into the behavior of solutions in diverse mathematical contexts.

Keywords:
calculus of variationsregularity

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Area of Science:

  • Mathematics
  • Mathematical Analysis

Background:

  • Variational problems are central to many areas of mathematics and physics.
  • Understanding the behavior of solutions, such as their monotonicity, is crucial for analysis.

Purpose of the Study:

  • To derive and discuss useful monotonicity formulae.
  • To explore these formulae in the context of various interesting variational problems.

Main Methods:

  • The paper is largely expository, focusing on the derivation process.
  • Detailed mathematical derivations of monotonicity properties are presented.

Main Results:

  • Novel monotonicity formulae are derived.
  • The applicability of these formulae to different variational problems is demonstrated.

Conclusions:

  • The derived monotonicity formulae provide valuable tools for analyzing variational problems.
  • This work contributes to a deeper understanding of solution behavior in mathematical analysis.