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VARIATIONAL EQUATIONS ON MIXED RIEMANNIANLORENTZIAN METRICS
, 2008
"... Abstract. A class of elliptichyperbolic 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 elliptichyperbolic 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 3space. This suggests an approach to generalized Plateau problems in 3dimensional spacetime via Hodge theory on the extended projective disc. Analogous variational problems arise on RiemannianLorentzian flow metrics in fiber bundles (twisted nonlinear Hodge equations), and on certain RiemannianLorentzian 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 3space, 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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