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A question about considering the solution of elliptic PDE with complex Laplacian as the critical point of a functional

I'm considering the elliptic PDE with complex Laplacian, for example, write $$ \Delta_c(\cdot):=-g^{i \bar{j}} \partial_i \partial_{\bar{j}}(\cdot), $$ and $$\Delta_c(u)=f,$$ by [P.Gauduchon, Math.Ann,...
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