A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations
type: Item
instance of: scholarly article
title: A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations
zbMATH Open document ID: 1072.35052
author: Cristian Rios
author: Eric T. Sawyer
author: R. L. Wheeden
published in: Advances in Mathematics
publication date: 2005-06-10T00:00:00Z
review text: The paper investigates the Monge-Ampère equation det \(D^2u=k(x,u,Du)\) in a convex domain of \(\mathbb R^n\), \(n\geq 3\), where \(k\) is a nonnegative smooth function. In case of \(k>0\) this equation is elliptic, and the theory is well understood. If \(k\) is permitted to vanish, the regularity of the solutions may fail to hold. To investigate the regularity of a solution \(u=u(x)\) when \(k\geq 0\), the following higher dimensional partial Legendre transform is introduced: \(s=x_1\), \(t_i=\frac{\partial u}{\partial x_1}(x)\), \(i=2,\dots, n\). Under such a transform, it is shown that the vector valued function \(\mathbf{ v} =(v_\ell)^n_{\ell=2}=(x_\ell(s,\mathbf{ t}))^n_{\ell=2}\) is a weak solution of the divergence form elliptic (degenerate) system \[ \Big\{\frac{\partial^2 }{\partial s^2}+\frac{\partial }{\partial \mathbf{ t'} }k\Big (co\Big[\frac{\partial \mathbf{ v} }{\partial \mathbf{ t'} }\Big]\Big)'\frac{\partial }{\partial \mathbf{ t} }\Big\}v_\ell=0,\;\;\;\ell=2,\dots,n, \] where \(\Big (co\Big[\frac{\partial \mathbf{ v} }{\partial \mathbf{ t'} }\Big]\Big)'\) denotes the transposed cofactor matrix of \(\frac{\partial \mathbf{ v} }{\partial \mathbf{t'}}\). The degeneracy term \(k\) appears now in the coefficient matrix. If \(k(x,u,Du)=(| x_1| ^{2m}+\psi(x))K(x,u,Du)\), with \(\psi\geq 0\), \(K>0\), \(m\) a positive integer and \(\psi^{\frac{1}{2m}}\) is Lipschitz continuous, and if det\(\Big[\frac{\partial^2 u }{\partial x_i\partial x_j}\Big]_{i,j=2}^n>0\), the theory of De Giorgi-Nash-Moser for subelliptic equations is used to derive regularity of \(u\). The theory applies in particular to the equation of prescribed Gaussian curvature det \(D^2 u=(1+| Du| ^2)^{\frac{n+2}{2}}k(x)\).
reviewed by: Giovanni Porru
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35H20
Mathematics Subject Classification ID: 35J70
zbMATH DE Number: 2174975
zbMATH Keywords: Monge-Ampère
zbMATH Keywords: partial Legendre, regularity
zbMATH Keywords: theory of De Giorgi-Nash-Moser
zbMATH Keywords: equation of prescribed Gaussian curvature
MaRDI profile type: Publication
full work available at URL: j.aim.2004.05.009
OpenAlex ID: W1968362061
cites work: The dirichlet problem for nonlinear second-order elliptic equations I. Monge-ampégre equation
cites work: Equazioni ellittiche del II\(^ \circ\) ordine e spazi \({\mathcal L}^{(2,\lambda)}\)
cites work: Q3900391
cites work: Interior estimates for elliptic systems of partial differential equations
cites work: Weighted Sobolev-Poincare Inequalities and Pointwise Estimates for a Class of Degenerate Elliptic Equations
cites work: Q3345173
cites work: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105)
cites work: Elliptic partial differential equations of second order
cites work: Regularity of a class of quasilinear degenerate elliptic equations
cites work: \(C^ 2\) a priori estimates for degenerate Monge-Ampère equations
cites work: Q4348142
cites work: On the Dirichlet problem for degenerate Monge-Ampère equations.
cites work: Isometric embeddings of surfaces with nonnegative curvature in \(\mathbb{R}^ 3\)
cites work: Q5519381
cites work: Q5829984
cites work: On Harnack's theorem for elliptic differential equations
cites work: Q5342626
cites work: A symbolic calculus for rough pseudodifferential operators
cites work: Regularity of degenerate Monge-Ampère and prescribed Gaussian curvature equations in two dimensions
cites work: Hölder continuity of weak solutions to subelliptic equations with rough coefficients
cites work: Regularity theory for quasilinear elliptic systems and Monge-Ampère equations in two dimensions
cites work: Pseudodifferential operators and nonlinear PDE
cites work: Q3752765
DOI: 10.1016/J.AIM.2004.05.009
description: scientific article; zbMATH DE number 2174975