Nirenberg's conjecture, proposed by Louis Nirenberg, concerns the behavior of solutions to certain nonlinear partial differential equations, particularly those related to elliptic equations. Specifically, the conjecture addresses the existence of solutions to the Dirichlet problem for certain elliptic equations involving the Laplacian operator with nonlinear boundary conditions. One of the key aspects of Nirenberg's conjecture is its relation to geometric properties, especially in the context of conformal geometry.
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