## 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

### Keywords:

elliptic regularity; almost complex 4-manifolds
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### References:

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