Properties of extremal functions for some nonlinear functionals on Dirichlet spaces

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.