Regularity of $p$-Harmonic Functions on the Plane

Higher-Order Regularity for the Solutions of Some Degenerate Quasilinear Elliptic Equations in the Plane

“…Unlike the known local regularity results for such equations, k is larger than 2 in many notable cases. These results generalize those in [13], which were established only for the p-Laplacian. Furthermore, local results are extended to obtain a global regularity result in some cases.…”

“…From this result it was pointed out in [21] that global W3'P(), C2'2/p-1(=), and WI-f-2/P,P(') regularity can indeed be achieved for solutions of the p-Laplacian in some physically relevant cases, though this result is only applicable to a limited class of boundary data. It should be noted that the results in [13] are sharp and such high-order regularity (local or global) is not generally true for the case p 5 0; see, for example, [17].…”

“…A first step in this direction was taken in [13], where C k, and W 2-Fk'q regularity has been established, using the theory of quasi-regular mappings for the solutions of the p-Laplacian in the plane with p 0, although the results are only local. Unlike most known results, k is larger than 2 in many interesting cases; for example, the p-Laplacian with 1 < p _< 2.…”

Full regularity of the free boundary in a Bernoulli-type problem in two dimensions

“…We also point out that in the case n = 2 Manfredi [Man88] showed that one can actually take α = μ. Interestingly, p-harmonic functions in the plane enjoy much higher regularity when p → 1, as shown by Iwaniec and Manfredi [IM89], however it is impossible to trace the constants in terms of u L ∞ (B 1 ) .…”

Non-uniqueness in a free boundary problem

“…The proof of Lemma 2 can be found in [4], [14] or [21] and in fact is true when B(w, 4r) ∩ Ω ⊂ R n . In R 2 the best Hölder exponent in Lemma 2 is known when p > 2 while for 1 < p ≤ 2 a solution has continuous second partials (see [12]). Proof.…”