We investigate a problem of approximation of a large class of nonlinear expressions$f$($x, u$, ∇$u$), including polyconvex functions. Here$u$: Ω→$R$^{$m$}, Ω⊂$R$^{$n$}, is a mapping from the Sobolev space$W$^{$1,p$}. In particular, when$p=n$, we obtain the approximation by mappings which are continuous, differentiable a.e. and, if in addition$n=m$, satisfy the Luzin condition. From the point of view of applications such mappings are almost as good as Lipschitz mappings. As far as we know, for the nonlinear problems that we consider, no natural approximation results were known so far. The results about the approximation of$f$($x, u$, ∇$u$) are consequences of the main result of the paper, Theorem 1.3, on a very strong approximation of Sobolev functions by locally weakly monotone functions.

We apply our model of quantum gravity to a Kerr-AdS spacetime of dimension $2 m+1$, $m\ge2$, where all rotational parameters are equal, resulting in a wave equation in a quantum spacetime which has a sequence of solutions that can be expressed as a product of stationary and temporal eigenfunctions. The stationary eigenfunctions can be interpreted as radiation and the temporal as gravitational waves. The event horizon corresponds in the quantum model to a Cauchy hypersurface that can be crossed by causal curves in both directions such that the information paradox does not occur. We also prove that the Kerr-AdS spacetime can be maximally extended by replacing in a generalized Boyer-Lindquist coordinate system the $r$ variable by $\rho=r^2$ such that the extended spacetime has a timelike curvature singularity in $\rho=-a^2$.

We give bounds for various homological invariants (including Castelnuovo-Mumford regularity, degrees of local cohomology, and injective dimension) of finitely generated VI-modules in the non-describing characteristic case. It turns out that the formulas of these bounds for VI-modules are the same as the formulas of corresponding bounds for FI-modules.

No abstract uploaded!

We show that, for all nonnegative integers k, Г, m and n, theYamabe invariant of #k(RP3)#Г(RP2 】 S1)#m(S2 】 S1)#n(S2. 】S1) is equal to the Yamabe invariant of RP3, provided k + Г ∶ 1. We then complete the classification (started by Bray and the second author) of all closed 3-manifolds with Yamabe invariant greater than that of RP3. More precisely, we show that such manifolds are either S3 or finite connected sums #m(S2 】 S1)#n(S2. 】S1),where S2. 】S1 is the nonorientable S2-bundle over S1. A key ingredient is Aubin’s Lemma [3], which says that if the Yamabe constant is positive, then it is strictly less than the Yam- abe constant of any of its non-trivial finite conformal coverings. This lemma, combined with inverse mean curvature flow and with analysis of the Green’s function for the conformal Laplacians on specific finite and normal infinite Riemannian coverings, will allow us to construct a family of nice test functions on the finite coverings and thus prove the desired result.