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VARIATIONAL EQUATIONS ON MIXED RIEMANNIAN-LORENTZIAN METRICS
, 2008
"... Abstract. A class of elliptic-hyperbolic equations is placed in the context of a geometric variational theory, in which the change of type is viewed as a change in the character of an underlying metric. A fundamental example of a metric which changes in this way is the extended projective disc, whic ..."
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Abstract. A class of elliptic-hyperbolic equations is placed in the context of a geometric variational theory, in which the change of type is viewed as a change in the character of an underlying metric. A fundamental example of a metric which changes in this way is the extended projective disc, which is Riemannian at ordinary points, Lorentzian at ideal points, and singular on the absolute. Harmonic fields on such a metric can be interpreted as the hodograph image of extremal surfaces in Minkowski 3-space. This suggests an approach to generalized Plateau problems in 3-dimensional space-time via Hodge theory on the extended projective disc. Analogous variational problems arise on Riemannian-Lorentzian flow metrics in fiber bundles (twisted nonlinear Hodge equations), and on certain Riemannian-Lorentzian manifolds which occur in relativity and quantum cosmology. The examples surveyed come with natural gauge theories and Hodge dualities. This paper is mainly a review, but some technical extensions are proven. MSC2000: 35M10, 53A10, 83C80 Key words: signature change, projective disc, Minkowski 3-space, equations of mixed type, nonlinear Hodge equations
POSITIVE SOLUTIONS OF ANISOTROPIC YAMABE–TYPE EQUATIONS IN R n
"... Abstract. We study entire positive solutions to the partial differential equation in Rn ∆xu + (α + 1) 2 |x | 2α ∆yu = −|x | 2α u n+2 n−2, where x ∈ R 2, y ∈ R n−2, n ≥ 3 and α> 0. We classify positive solutions with second order derivatives satisfying a suitable growth near the set x = 0. 1. ..."
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Abstract. We study entire positive solutions to the partial differential equation in Rn ∆xu + (α + 1) 2 |x | 2α ∆yu = −|x | 2α u n+2 n−2, where x ∈ R 2, y ∈ R n−2, n ≥ 3 and α> 0. We classify positive solutions with second order derivatives satisfying a suitable growth near the set x = 0. 1.
L² estimates for the eigenfunctions corresponding to real eigenvalues of the Tricomi operator
, 2009
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