Hardy Type Inequalities for Δλ\Delta_\lambda-Laplacians

We derive Hardy type inequalities for a large class of sub-elliptic operators that belong to the class of $\Delta_\lambda$-Laplacians and find explicit values for the constants involved. Our results generalize previous inequalities obtained for Grushin type operators $$\Delta_{x}+ |x|^{2\alpha}\Delta_{y},\qquad\ (x,y)\in\mathbb{R}^{N_1}\times\mathbb{R}^{N_2},\ \alpha\geq 0,$$ which were proved to be sharp

Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications

This article is a continuation of our previous work [5], where we formulated general existence theorems for pullback exponential attractors for asymptotically compact evolution processes in Banach spaces and discussed its implications in the autonomous case. We now study properties of the attractors and use our theoretical results to prove the existence of pullback exponential attractors in two examples, where previous results do not apply

Pullback exponential attractors for evolution processes in banach spaces: Theoretical results

We construct exponential pullback attractors for time continuous asymptotically compact evolution processes in Banach spaces and derive estimates on the fractal dimension of the attractors. We also discuss the corresponding results for autonomous processes

Attractors for a class of semi-linear degenerate parabolic equations

We consider degenerate parabolic equations of the form ∂tu = Δλu + f(u) u{pipe}∂Ω = 0, u{pipe}t=0 = u0 in a bounded domain Ω ⊂N, where Δλ is a subelliptic operator of the type (Formula presented.) We prove global existence of solutions and characterize their longtime behavior. In particular, we show the existence and finite fractal dimension of the global attractor of the generated semigroup and the convergence of solutions to an equilibrium solution when time tends to infinity

On the positivity of solutions of systems of stochastic PDEs

We study the positivity of solutions of a class of semi-linear parabolic systems of stochastic partial differential equations by considering random approximations. For the family of random approximations we derive explicit necessary and sufficient conditions such that the solutions preserve positivity. These conditions imply the positivity of the solutions of the stochastic system for both Itô's and Stratonovich's interpretation of stochastic differential equations. We study the positivity of solutions of a class of semi-linear parabolic systems of stochastic partial differential equations by considering random approximations. For the family of random approximations we derive explicit necessary and sufficient conditions such that the solutions preserve positivity. These conditions imply the positivity of the solutions of the stochastic system for both Itô's and Stratonovich's interpretation of stochastic differential equations

Domain-wall/Cosmology correspondence in adS/dS supergravity

We realize the domain-wall/cosmology correspondence for (pseudo)supersymmetric domain walls (cosmologies) in the context of four-dimensional supergravity. The OSp(2|4)-invariant anti-de Sitter (adS) vacuum of a particular N=2 Maxwell-Einstein supergravity theory is shown to correspond to the OSp(2^*|2,2)-invariant de Sitter (dS) vacuum of a particular pseudo-supergravity model, with `twisted' reality conditions on spinors. More generally, supersymmetric domain walls of the former model correspond to pseudo-supersymmetric cosmologies of the latter model, with time-dependent pseudo-Killing spinors that we give explicitly.Comment: 21 pages;v2: rewritten to clarify the link with fake supergravity -- main results unchanged; v3: typos corrected, two refs added, JHEP versio

Janus and Multifaced Supersymmetric Theories

We investigate the various properties Janus supersymmetric Yang-Mills theories. A novel vacuum structure is found and BPS monopoles and dyons are studied. Less supersymmetric Janus theories found before are derived by a simpler method. In addition, we find the supersymmetric theories when the coupling constant depends on two and three spatial coordinates.Comment: 20 pages, no figures, typos, equations corrected. Additional comment

Pseudo-supersymmetry and a Tale of Alternate Realities

We discuss how all variant 10d and 11d maximal supergravities, including star supergravities and supergravities in different signatures, can be obtained as different real slices of three complex actions. As an application we study the recently introduced domain-wall/cosmology correspondence in this approach. We give an example in 9d and 10d where the domain-wall and corresponding cosmology can be viewed as different real slices of the same complex solution. We argue how in this case the pseudo-supersymmetry of the cosmological solutions can be understood as the invariance under supersymmetry of a variant supergravity.Comment: 32 page