A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations
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 \(D^2 u=(1+| Du| ^2)^{\frac{n+2}{2}}k(x)\).
equation of prescribed Gaussian curvature
scientific article; zbMATH DE number 2174975
scientific article