This paper demonstrates the existence of harmonic functions in m-convex domains, highlighting their unique properties and implications.
The well-known classical Dirichlet problem states that if D⊂ Rⁿ is a regular domain, then for any continuous function φ (ξ )∈ C(∂ D), there exists a unique harmonic function ω (x)∈ C(D̄), such that ω |∂ D =φ. In the work of Sadullaev-Sharipov, under an additional condition of strict m-convexity of the domain D⊂ Rⁿ, an analogous result for m-convex $(m-cv)$ functions has been proven.
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Руслан Шарипов (2025) studied this question.
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