Super regularity for Beltrami systems
We prove a surprising higher regularity for solutions to the nonlinear elliptic autonomous Beltrami equation in a planar domain \(Ω\), \[f_z = A(f_z) a.e. zΩ,\] when \(A\) is linear at \(∞\). Namely \(W^1,1_loc (Ω)\) solutions are \(W^2,2+ε_loc (Ω)\). Here \(ε>0\) depends explicitly on the ellipticity bounds of \(A\). The condition "is linear at \(∞\)" is necessary - the result is false for the equation \(f_z = k|f_z|\), for any \(0<k<1\), (\(k=0\) is Weyl’s lemma) and the improved regularity is sharp, but can be further improved if, for instance, \(A\) is smooth. We also discuss the subsequent higher regularity implications for fully non-linear Beltrami systems \[f_z = A(z, f_z) a.e. zΩ.\] There the condition "linear at \(∞\)" also implies improved regularity for \(W^1,1_loc (Ω)\) solutions. , number=1, journal=Annales Fennici Mathematici, author=Martin, Gaven J., year=2021, month=Jun., pages=59–65