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Continuous solutions of nonlinear Cauchy-Riemann equations and pseudoholomorphic curves in normal coordinates. (English) Zbl 1375.35152

The paper establishes elliptic regularity of solutions to Cauchy-Riemann equations of the form \(\frac{\partial u}{\partial\overline{z}}=E(z,u)\) under weak assumptions. In the linear case, \(E(z,u)=P(z)\), \(u\) is assumed to be continuous with partial derivatives defined everywhere except at countably many points, and \(P\in L^{p}_{loc}\), \(p\geq2\). The result is that \(u\) is then locally in \(W^{1,2}\), and in \(C^{0,\alpha}\) with \(\alpha=1-\frac{p}2\) when \(p>2\). Similar regularity results for the non-linear equation with continuous or Hölder continuous \(E\) follow as corollaries. A counterexample is given to show that a solution to the linear equation with continuous \(P\) and \(u\) having partial derivatives everywhere may have locally unbounded \(\frac{\partial u}{\partial\overline{z}}\).
In the separable case, \(E(z,u)=f(u)g(z)\), explicit and implicit local formulas for solutions are derived without a priori regularity assumptions. The results are then used to derive coordinate formulas for \(J\)-holomorphic curves in some almost complex \(4\)-manifolds, and to show that a continuous almost complex structure can admit differentiable \(J\)-holomorphic curves that are not \(C^1\).

MSC:

35J46 First-order elliptic systems
30G20 Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.)
32Q65 Pseudoholomorphic curves
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