A Classification of Isolated Singularities of Elliptic Monge‐Ampére Equations in Dimension Two. Issue 12 (6th May 2015)

• Record Type: Journal Article
• Authors: Gálvez, José A.; Jiménez, Asun; Mira, Pablo
• Is Part Of: Communications on pure and applied mathematics. Volume 68:Issue 12(2015:Dec.)
• Page Start: 2085
• Page End: 2107
• Publication Date: 2015-05-06
• Subjects: Mathematics -- Periodicals; Mechanics -- Periodicals; Mathématiques -- Périodiques; Mécanique -- Périodiques; 510.5
• DOI: 10.1002/cpa.21581
• Languages: English
• ISSNs: 0010-3640

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Let ℳ<sub>1</sub> denote the space of solutions <italic>z</italic>(<italic>x</italic>, <italic>y</italic>) to an elliptic, real analytic Monge‐Ampére equation <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrn9n1q9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00103640:media:cpa21581:cpa21581-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>det</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>D</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>φ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>, </mml:mo><mml:mi>y</mml:mi><mml:mo>, </mml:mo><mml:mi>z</mml:mi><mml:mo>, </mml:mo><mml:mi>D</mml:mi><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula> whose graphs have a non‐removable isolated singularity at the origin. We prove that ℳ<sub>1</sub> is in one‐to‐one correspondence with ℳ<sub>2</sub> × ℤ<sub>2</sub>, where ℳ<sub>2</sub> is a suitable subset of the class of regular, real analytic, strictly convex Jordan curves in ℝ<sub>2</sub>. We also describe the asymptotic behavior of solutions of the Monge‐Ampére equation in the <italic>C</italic><sup><italic>k</italic></sup>‐smooth case, and a general existence theorem for isolated singularities of analytic solutions of the more general equation <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrn9n1rt" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00103640:media:cpa21581:cpa21581-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>det</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>D</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>z</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="script">A</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>, </mml:mo><mml:mi>y</mml:mi><mml:mo>, </mml:mo><mml:mi>z</mml:mi><mml:mo>, </mml:mo><mml:mi>D</mml:mi><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>φ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>, </mml:mo><mml:mi>y</mml:mi><mml:mo>, </mml:mo><mml:mi>z</mml:mi><mml:mo>, </mml:mo><mml:mi>D</mml:mi><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>.© 2015 Wiley Periodicals, Inc.</p> </abstract>