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Dirichlet problem for the equation of a given Lorentz-Gaussian curvature. (English. Russian original) Zbl 0724.35039

Ukr. Math. J. 42, No. 12, 1538-1545 (1990); translation from Ukr. Mat. Zh. 42, No. 12, 1704-1710 (1990).
Let \(\Omega\) be a bounded, strictly convex smooth domain in \(R^ n\) and let \(K\phi =\det (D^ 2\phi)/(1-| D\phi |^ 2)^{(n+2)/2}\). Denote by \(K_+\) the set of all strictly convex functions \(\phi\) in \(\Omega\) such that \(| d\phi | <1\) in \({\bar \Omega}\). It is shown that for every f in \(K_+\cap C^{\infty}(\Omega)\) and every \(\nu\) in \(C^{\infty}({\bar \Omega})\), \(\nu >0\), the problem \(K\phi =f\), \(\phi |_{\partial \Omega}=\nu |_{\partial \Omega}\) has a unique solution in \(K_+\cap C^{\infty}(\Omega)\). The problem has a geometric interpretation as indicated in the title of the paper.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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