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Properties of extremal functions for some nonlinear functionals on Dirichlet spaces

Authors: Alec Matheson and Alexander R. Pruss
Journal: Trans. Amer. Math. Soc. 348 (1996), 2901-2930
MSC (1991): Primary 30A10, 30D99; Secondary 28A20, 49J45, 49K99
MathSciNet review: 1357401
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Abstract: Let $ \mathfrak {B}$ be the set of holomorphic functions $f$ on the unit disc $D$ with $f(0)=0$ and Dirichlet integral $(1/\pi ) \iint _{D} |f'|^{2}$ not exceeding one, and let $ \mathfrak {b}$ be the set of complex-valued harmonic functions $f$ on the unit disc with $f(0)=0$ and Dirichlet integral $(1/2)(1/\pi ) \iint _{D} |\nabla f|^{2}$ not exceeding one. For a (semi)continuous function $\Phi :[0,\infty ) \to \mathbb {R}$, define the nonlinear functional $\Lambda _{\Phi }$ on $ \mathfrak {B}$ or $ \mathfrak {b}$ by $\Lambda _{\Phi }(f)={\frac {1}{2\pi }} \int _{0}^{2\pi }\Phi (|f(e^{i\theta })|)\,d\theta $. We study the existence and regularity of extremal functions for these functionals, as well as the weak semicontinuity properties of the functionals. We also state a number of open problems.

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Additional Information

Alec Matheson
Affiliation: Department of Mathematics, Lamar University, Beaumont, Texas 77710

Alexander R. Pruss
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2

Keywords: Dirichlet space, Dirichlet integral, Beurling's estimate, convergence in measure, Chang-Marshall inequality, harmonic majorants and rearrangement, optimization problems, necessary conditions for extremality, regularity of extremals
Received by editor(s): September 8, 1994
Received by editor(s) in revised form: September 5, 1995
Additional Notes: The research of the second author was partially supported by Professor J. J. F. Fournier’s NSERC Grant #4822. Portions of this paper also appear in a part of the second author’s doctoral dissertation.
Article copyright: © Copyright 1996 American Mathematical Society

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