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Functions monotone close to boundary. (English) Zbl 1099.31004

A continuous function \(f : D \rightarrow \mathbb{R}\) is called monotone close to \(\Gamma\) if for every subdomain \(\Delta \subset D\) with \(\partial^{\prime\prime} \Delta \subset \Gamma\) the inequality \[ \text{osc}\, (f, \Delta) \leq \text{osc}\, (f, \partial^{\prime} \Delta) \] holds, where \(\partial^{\prime} \Delta = \tilde{\partial}\, \Delta \setminus \tilde{\partial}\, D\), \(\partial^{\prime\prime} \Delta = \tilde{\partial}\, \Delta \cap \tilde{\partial}\, D\). The following characterization of functions monotone close to the boundary is obtained: Let \(D\) be a subdomain of \(\mathbb{R}^2\), \(\Gamma \subset \partial D\) be an open Jordan arc, and \(f\in ACL^{\Phi}(D)\). If \(f\) is close to \(\Gamma\), then for every pair of points \(a, b \;D\) with \[ \rho_D(a, b) < \delta_D(a, b; \Gamma), \] the estimate
\[ | f(a) - f(b)| \leq \kappa_{0}(\rho_D(a, b); \delta_D(a, b; \Gamma), \Phi, I), \] holds where \[ I = \int_{D} \Phi(| \nabla\, f| \,d x_{1} \,d x_{2}, \]
\[ \kappa_{0}(\epsilon; \epsilon_{0}; \Phi; I) = \sup\left\{\kappa: \int_{\epsilon}^{\epsilon_0} \Phi\left(\frac{\kappa}{2\pi \tau}\right) \tau \,d \tau \leq \frac{1}{2\pi} I\right\}. \] Sufficient conditions for monotonicity close to the boundary are given illustrated by certain prominent examples.

MSC:

31A20 Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
31C45 Other generalizations (nonlinear potential theory, etc.)
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