Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation

Let Ω be a smooth bounded domain in , with N?5, a>0, α?0 and . We show that the exponent plays a critical role regarding the existence of least energy (or ground state) solutions of the Neumann problem

     

Geodesic currents and Teichmüller space

Consider a hyperbolic surface X of infinite area. The Liouville map assigns to any quasiconformal deformation of X a measure on the space of geodesics of the universal covering X? of X. We show that the Liouville map is a homeomorphism from the Teichmüller space onto its image, and that the image is closed and unbounded. The set of asymptotic rays to consists of all bounded measured laminations on X. Hence, the set of projective bounded measured laminations is a natural boundary for . The action of the quasiconformal mapping class group on continuously extends to this boundary for .     

An analytic characterization of the eigenvalues of self-adjoint extensions

Let be a self-adjoint extension in of a fixed symmetric operator A in . An analytic characterization of the eigenvalues of is given in terms of the Q-function and the parameter function in the Krein-Naimark formula. Here K and are Krein spaces and it is assumed that locally has the same spectral properties as a self-adjoint operator in a Pontryagin space. The general results are applied to a class of boundary value problems with λ-dependent boundary conditions.     

Holomorphy of spectral multipliers of the Ornstein-Uhlenbeck operator

Let γ be the Gauss measure on and the Ornstein-Uhlenbeck operator. For every p in [1,∞)?{2}, set , and consider the sector . The main results of this paper are the following. If p is in (1,∞)?{2}, and , i.e., if M is an Lp(γ)uniform spectral multiplier of in our terminology, and M is continuous on , then M extends to a bounded holomorphic function on the sector . Furthermore, if p=1 a spectral multiplier M, continuous on , satisfies the condition if and only if M extends to a bounded holomorphic function on the right half-plane, and its boundary value M(i·) on the imaginary axis is the Euclidean Fourier transform of a finite Borel measure on the real line. We prove similar results for uniform spectral multipliers of second order elliptic differential operators in divergence form on belonging to a wide class, which contains . From these results we deduce that operators in this class do not admit an H^∞ functional calculus in sectors smaller than .     

Malliavin calculus in abstract Wiener space using infinitesimals

Using infinitesimals, we develop Malliavin calculus on spaces which result from the classical Wiener space by replacing with any abstract Wiener space .We start from a Brownian motion b on a Loeb probability space Ω with values in the Banach space is the standard part of a ∗finite-dimensional Brownian motion B. Then we define iterated Itô integrals as standard parts of internal iterated Itô integrals. The integrator of the internal integrals is B and the values of the integrands are multilinear forms on , where is a ∗finite-dimensional linear space over between the Hilbert space and its ∗-extension .In the first part we prove a chaos decomposition theorem for L²-functionals on Ω that are measurable with respect to the σ-algebra generated by b. This result yields a chaos decomposition of L²-functionals with respect to the Wiener measure on the standard space of -valued continuous functions on [0,1]. In the second part we define the Malliavin derivative and the Skorohod integral as standard parts of internal operators defined on ∗finite-dimensional spaces. In an application we use the transformation rule for finite-dimensional Euclidean spaces to study time anticipating and non-anticipating shifts of Brownian motion by Bochner integrals (Girsanov transformations).     

Exponentiable monomorphisms in categories of domains

We give a characterization of exponentiable monomorphisms in the categories of ω-complete posets, of directed complete posets and of continuous directed complete posets as those monotone maps f that are convex and that lift an element (and then a queue) of any directed set (ω-chain in the case of ) whose supremum is in the image of f (Theorem 1.9). Using this characterization, we obtain that a monomorphism f:X→B in (, ) exponentiable in w.r.t. the Scott topology is exponentiable also in (, ). We prove that the converse is true in the category , but neither in , nor in .