Solutions of some Monge-Ampère equations with isolated and line singularities

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• 作者：Tianling Jin^a ; ^b ; ^{tianlingjin@ust.hk" class="auth_mail" title="E-mail the corresponding author} ; ^{tianling@caltech.edu" class="auth_mail" title="E-mail the corresponding author} ; Jingang Xiong^c ; ¹ ; ^{jx@bnu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35J96 ; 35B40 ; 35B53 ; 35J67 ; 35N25
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：5 February 2016
• 年：2016
• 卷：289
• 期：Complete
• 页码：114-141
• 全文大小：492 K

文摘

In this paper, we study existence, regularity, classification, and asymptotic behaviors of solutions of some Monge–Ampère equations with isolated and line singularities. We classify all solutions of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815004946&_mathId=si1.gif&_user=111111111&_pii=S0001870815004946&_rdoc=1&_issn=00018708&md5=ed0928b76712007f6b5c1b19333cce39" title="Click to view the MathML source">det⁡∇²u=1class="mathContainer hidden">class="mathCode">$\det {\mathrm{\nabla }}^{2}u=1$ in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815004946&_mathId=si2.gif&_user=111111111&_pii=S0001870815004946&_rdoc=1&_issn=00018708&md5=d006828ae6f9bd7ab39c2fc2e128b2a4" title="Click to view the MathML source">Rⁿclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^n$ with one puncture point. This can be applied to characterize ellipsoids, in the same spirit of Serrin's overdetermined problem for the Laplace operator. In the case of having k   non-removable singular points for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815004946&_mathId=si3.gif&_user=111111111&_pii=S0001870815004946&_rdoc=1&_issn=00018708&md5=b38be173576e3f6996dcc24f2f15b9b7" title="Click to view the MathML source">k>1class="mathContainer hidden">class="mathCode">$k>1$, modulo affine equivalence the set of all generalized solutions can be identified as an explicit orbifold. We also establish existence of global solutions with general singular sets, regularity properties, and optimal estimates of the second order derivatives of generalized solutions near the singularity consisting of a point or a straight line. The geometric motivation comes from singular semi-flat Calabi–Yau metrics.