A Classification of Isolated Singularities of Elliptic Monge‐Ampére Equations in Dimension Two. Issue 12 (6th May 2015) - Gálvez, José A.; Jiménez, Asun; Mira, Pablo
Abstract Let ℳ1 denote the space of solutions z(x, y) to an elliptic, real analytic Monge‐Ampére equation <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrn9n1q9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />det(D2z)=φ(x, y, z, Dz)>0</alternatives> whose graphs have a non‐removable isolated singularity at the origin. We prove that ℳ1 is in one‐to‐one correspondence with ℳ2 × ℤ2, where ℳ2 is a suitable subset of the class of regular, real analytic, strictly convex Jordan curves in ℝ2. We also describe the asymptotic behavior of solutions of the Monge‐Ampére equation in the Ck‐smooth case, and a general existence theorem for isolated singularities of analytic solutions of the more general equation <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrn9n1rt" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />det(D2z+A(x, y, z, Dz))=φ(x, y, z, Dz)>0</alternatives>.© 2015 Wiley Periodicals, Inc.