To the theory of variational method for Beltrami equations (Q690516)

The authors extend the variational method for the Beltrami equation \(f_{\bar z}=\mu f_z\) (\(\|\mu\|_\infty=k<1\)) to the setting of degenerate Beltrami equations, which allows for \(\|\mu\|_\infty=1\). In the non-degenerate case it is well-known that extremal maps tend to have constant value of \(|\mu|\), which denotes the modulus of the complex dilatation. The degenerate case is quite different because the constraint on \(\mu\) may have an integral form, e.g., \(\int_{\mathbb C} \Psi(|\mu|)\leq 1\), instead of pointwise \(|\mu|\leq k\) a.e. In particular, one cannot expect extremal maps to have constant \(|\mu|\). The authors prove that for a wide range of variational problems, any extremal map will attain equality in the integral dilatation constraint \(\int_{\mathbb C} \Psi(|\mu|) = 1\). Also, if the constraint has the form of a differential inclusion, \(\mu(z)\in M(z)\subset \{z:|z|<1\}\), then an extremal map satisfies \(\mu(z)\in \partial M(z)\) almost everywhere.