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Properties of extremal functions for some nonlinear functionals on Dirichlet spaces

Authors:	Alec Matheson Alexander R Pruss

Institution:	Department of Mathematics, Lamar University, Beaumont, Texas 77710 ; Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2

Abstract:	Let $\mathfrak {B}$ be the set of holomorphic functions on the unit disc with and Dirichlet integral $(1/\pi ) \iint _{D} \|f'\|^{2}$ not exceeding one, and let $\mathfrak {b}$ be the set of complex-valued harmonic functions on the unit disc with 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.

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

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