Two Necessary and Sufficient Conditions to the Solvability of the Exterior Dirichlet Problem for the Monge-Ampère Equation
arxiv(2024)
摘要
The present paper provides two necessary and sufficient conditions for the
existence of solutions to the exterior Dirichlet problem of the Monge-Ampère
equation with prescribed asymptotic behavior at infinity. By an adapted smooth
approximation argument, we prove that the problem is solvable if and only if
the boundary value is semi-convex with respect to the inner boundary, which is
our first proposed new concept. Along the lines of Perron's method for Laplace
equation, we obtain the threshold for solvability in the asymptotic behavior at
infinity of the solution, and remove the C^2 regularity assumptions on the
boundary value and on the inner boundary which are required in the proofs of
the corresponding existence theorems in the recent literatures.
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