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The degenerate Monge-Ampère equations with the Neumann condition. (English) Zbl 1475.35172

Suppose \(\Omega\subset\mathbb{R}^n\) is a \(C^{3,1}\), bounded and uniformly convex domain and let \(\varphi\in C^{1,1}(\bar{\Omega}\times\mathbb{R})\), where \(\varphi=\varphi(x,z)\) satisfies \(\varphi_z\geq\gamma_0\) on \(\partial\Omega\times\mathbb{R}\) for some positive constant \(\gamma_0\). The authors establish existence and uniqueness of a convex solution \(u\in C^{1,1}(\bar{\Omega})\) to the Monge-Ampére equation \(\mathrm{det}D^2u=f\) in \(\Omega\), together with the Neumann boundary value condition \(D_\nu u=\varphi(x,u)\) on \(\partial\Omega\). The data \(f\) is merely assumed to be nonnegative and such that the optimal exponent condition \(f^{\frac{1}{n-1}}\in C^{1,1}(\bar{\Omega})\) is satisfied (some extra condition is required on \(f\) in case it vanishes at a boundary point). To do so, a priori second order derivative estimates are obtained for sufficiently smooth solutions.

MSC:

35J96 Monge-Ampère equations
35J25 Boundary value problems for second-order elliptic equations
35J70 Degenerate elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
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