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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
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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.

Variational analysis for a nonlinear elliptic problem on theSierpiński gasket
Gabriele Bonanno, Giovanni Molica Bisci, Vicenţiu Rădulescu
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 18 / Issue 4 / October 2012

Published online by Cambridge University Press:

16 January 2012, pp. 941-953

Print publication:

October 2012
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Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinearterm, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem onthe Sierpiński gasket is proved. Our approach is based on variational methods and on someanalytic and geometrical properties of the Sierpiński fractal. The abstract results areillustrated by explicit examples.

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
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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.