Lipschitz Continuity for Harmonic Functions and Solutions of the α¯-Poisson Equation

Mateljević, Miodrag; Mutavdžić, Nikola; Khalfallah, Adel

doi:10.3390/axioms12100998

Axioms

2023

DOI: 10.3390/axioms12100998

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Lipschitz Continuity for Harmonic Functions and Solutions of the α¯-Poisson Equation

Miodrag Mateljević,

Nikola Mutavdžić,

Adel Khalfallah

Abstract: In this paper we investigate the solutions of the so-called α¯-Poisson equation in the complex plane. In particular, we will give sufficient conditions for Lipschitz continuity of such solutions. We also review some recently obtained results. As a corollary, we can restate results for harmonic and (p,q)-harmonic functions.

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2024

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Cited by 2 publications

References 21 publications

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Schwarz Type Lemmas for Generalized Harmonic Functions

Khalfallah,

Mhamdi

2024

Bull. Malays. Math. Sci. Soc.

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Schwarz Type Lemmas for Generalized Harmonic Functions

Khalfallah,

Mhamdi

2024

Bull. Malays. Math. Sci. Soc.

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Interpolation formulas for 1-harmonic functions on the unit circle

Djukic,

Mutavdzic,

Mutavdzic-Djukic

2024

Filomat

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Ageneralization of the deeply investigated harmonic functions, known as ?-harmonic functions, have recently gained considerable attention. Similarly to the harmonic functions, an ?-harmonic function u on the unit disc D is uniquely determined by its values on the boundary of the disc ?D. In fact, for any z ? D, the value of u(z) can be given as a contour integral over ?D with a modified Poisson kernel. However, this integral can be difficult to evaluate, or the values on the boundary are known only empirically. In such cases, approximating u(z) with an interpolatory formula, as a weighted sum of values of u at n nodes on ?D, can be an attractive alternative. The nodes and weights are to be chosen so that the degree d of exactness of the formula is maximized. In other words, the formula should be exact for all basis functions for ?-harmonic functions of degree up to d, with d as large as possible. In the case of harmonic functions, it is known that there is an interpolation formula of degree of exactness as large as d = n ? 1. The objective of this paper are formulas of this type for ?-harmonic functions. We will prove that, given n, in this case the degree of exactness cannot be n ? 1, but there is a unique interpolation formula of degree n ? 2. Finally, we will prove convergence of such formulas to u(z) as n ? ?.

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Lipschitz Continuity for Harmonic Functions and Solutions of the α¯-Poisson Equation

Cited by 2 publications

References 21 publications

Schwarz Type Lemmas for Generalized Harmonic Functions

Schwarz Type Lemmas for Generalized Harmonic Functions

Interpolation formulas for 1-harmonic functions on the unit circle

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