On the Boundary Behavior of Positive Solutions of Elliptic Differential Equations

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Let u be a positive harmonic function in the unit ball B₁ ⊂ Rⁿ and let μ be the boundary measure of u. Consider a point x∈ ∂ B₁ and let n(x) denote the unit normal vector at x. Let α be a number in (-1,n-1] and A ∈ [0,+∞). We prove that u(x+n(x)t)tα → A as t → +0 if and only if μ(Bᵣ(x))rⁿ⁻¹ rα → CαA as r→+0, where Cα= πⁿ/²Γ(n-α+1/2)Γ(α+1/2). For α=0 it follows from the theorems by Rudin and Loomis which claim that a positive harmonic function has a limit along the normal iff the boundary measure has the derivative at the corresponding point of the boundary. For α=n-1 it concerns about the point mass of μ at x and it follows from the Beurling minimal principle. For the general case of α∈ (-1,n-1) we prove it with the help of the Wiener Tauberian theorem in a similar way to Rudin's approach. Unfortunately this approach works for a ball or a half-space only but not for a general kind of domain. In dimension 2 one can use conformal mappings and generalise the statement above to sufficiently smooth domains, in dimension n≥ 3 we showed that this generalisation is possible for α∈ [0,n-1] due to harmonic measure estimates. The last method leads to an extension of the theorems by Loomis, Ramey and Ullrich on non-tangential limits of harmonic functions to positive solutions of elliptic differential equations with Holder continuous coefficients.