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3 results

  • 9 - The ⋆-Function

  • Albert Baernstein II, Washington University, St Louis
  • Book:
    Symmetrization in Analysis
    Published online:
    22 February 2019
    Print publication:
    14 March 2019, pp 299-355

  • Summary

    This chapter marks the debut of the star function in the book. Each type of rearrangement has an associated star function, which is an indefinite integral of the rearranged function. This chapter proves ``subharmonicity'' theorems for the star function, expressing the fact that if a function satisfies a Poisson-type partial differential equation then its star function satisfies a related differential inequality. In the simplest case of circular symmetrization in the plane, the result says that if a function is subharmonic then so is its star function. Subharmonicity is applied in the succeeding chapters to yield comparison theorems for solutions of partial differential equations and extremal results in complex analysis.

  • TWO NONTRIVIAL WEAK SOLUTIONS FOR THE DIRICHLET PROBLEM ON THE SIERPIŃSKI GASKET

  • Part of
  • DENISA STANCU-DUMITRU
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 85 / Issue 3 / June 2012
    Published online by Cambridge University Press:
    12 December 2011, pp. 395-414
    Print publication:
    June 2012
    • Article
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  • We study a Dirichlet problem involving the weak Laplacian on the Sierpiński gasket, and we prove the existence of at least two distinct nontrivial weak solutions using Ekeland’s Variational Principle and standard tools in critical point theory combined with corresponding variational techniques.