Let Ω be a bounded domain in Rn, n ≥ 3 with smooth boundary ∂Ω and a small hole. We give the first example of sign-changing bubbling solutions to the nonlinear elliptic problem −∆u = |u| n+2 n−2+ε−1u in Ω, u = 0 on ∂Ω, where ε is a small positive parameter. The basic cell in the construction is the signchanging nodal solution to the critical Yamabe problem −∆w = |w| 4 n−2w, w ∈ D(R) which has l...

The topological sensitivity analysis consists in studying the behavior of a given shape functional when the topology of the domain is perturbed, typically by the nucleation of a small hole. This notion forms the basic ingredient of different topology optimization / reconstruction algorithms. From the theoretical viewpoint, the expression of the topological sensitivity is wellestablished in many...

Over the past few years the understanding of the microscopic theory of black hole entropy has made important conceptual progress by recognizing that the degeneracies are encoded in partition functions which are determined by higher rank automorphic representations, in particular in the context of Siegel modular forms of genus two. In this brief review some of the elements of this framework are ...

The topological sensitivity analysis consists in studying the behavior of a given shape functional when the topology of the domain is perturbed, typically by the nucleation of a small hole. This notion forms the basic ingredient of different topology optimization / reconstruction algorithms. From the theoretical viewpoint, the expression of the topological sensitivity is wellestablished in many...

Vafa’s (11+1) F theory is extended by means of Bars’ 2T holographic theory to yield a 14d Multiverse theory that permeates the brane of a 12d Universe in which both the Universe and the Multiverse have (3+1) spacetimes. Given the 2d toroidal compactification of F theory, we conjecture that the Multiverse has a 4d Cartesian compactification that is filled with 3D+T spacetime via the standard 6d ...

Elliptic problems in a two-dimensional domain containing one or several small holes is studied asymptotically in the limit of small hole radius ". As a result of the perturbation , the asymptotic solution to this class of problems contains an innnite logarithmic expansion in powers of ?1= log ". A hybrid asymptotic-numerical technique is used to sum these innnite logarithmic expansions and obta...

Given a compact domain of a 3-dimensional hypersurface on a vacuum spacetime, a scalar (the “non-Kerrness”) is constructed by solving a Dirichlet problem for a second order elliptic system. If such scalar vanishes, and a set of conditions are satisfied at a point, then the domain of dependence of the compact domain is isometric to a portion of a member of the Kerr family of solutions to the Ein...

Relativistic radiative transfer problems require the calculation of photon trajectories in curved spacetime. We present a novel technique for rapid and accurate calculation of null geodesics in the Kerr metric. The equations of motion from the Hamilton-Jacobi equation are reduced directly to Carlson’s elliptic integrals, simplifying algebraic manipulations and allowing all coordinates to be com...

We propose a microscopic CFT description of magnetically charged black holes in IIA compactifications on elliptic fibrations based on the Fourier-Mukai transform. The physical derivation of this model involves a chain of string duality transformations including the 4D/5D black hole correspondence. We compute the asymptotic behavior of the microstate degeneracy in a certain limit of large charge...

We consider the elliptic problem ∆u+up = 0, u > 0 in an exterior domain, Ω = RN \D under zero Dirichlet and vanishing conditions, where D is smooth and bounded, and p is supercritical, namely p > N+2 N−2 . We prove that the associated Dirichlet problem has infinitely many positive solutions with slow decay O(|x| 2 p−1 ) at infinity. In addition, a fast decay solution exists if p is close enough...