A Liouville Theorem for Solutions of Degenerate Monge-Ampere Equations

作者：Jin Tianling; Xiong Jingang^*

来源：Communications in Partial Differential Equations, 2014, 39(2): 306-320.

DOI：10.1080/03605302.2013.814143

• Degenerate Monge-Ampere equations
• Grushin type equations
• Liouville theorem
• Primary 35J96
• Secondary 35J70

In this paper, we give a new proof of a celebrated theorem of Jorgens which states that every classical convex solution of
det del(2)u(x) = 1 in R-2
has to be a second order polynomial. Our arguments do not use complex analysis, and can be applied to establish such Liouville type theorems for solutions of a class of degenerate Monge-Ampere equations. We prove that every convex generalized (or Alexandrov) solution of
det del(2)u(x(1), x(2)) = vertical bar x(1)vertical bar(alpha) in R-2,
where alpha > -1, has to be
u(x(1), x(2)) = alpha/(alpha + 2)(alpha + 1)vertical bar x(1)vertical bar(2+alpha) + ab(2)/2x(1)(2) + bx(1)x(2) + 1/2ax(2)(2) + l(x(1), x(2))
for some constants a>0, b and a linear function l(x(1), x(2)). This work is motivated by the Weyl problem with nonnegative Gauss curvature.

• 出版日期2014-2-1
• 单位北京大学