We study the first eigenpair of a Dirichlet spectral problem for singularly perturbed convection-diffusion operators with oscillating locally periodic coefficients. It follows from the results of [A. Piatnitski and V. Rybalko, On the first eigenpair of singularly perturbed operators with oscillating coefficients. Preprint www.arxiv.org, arXiv:1206.3754] that the first eigenvalue remains bounded only if the integral curves of the so-called effective drift have a nonempty ω-limit set. Here we consider...

This text aims to describe results of the authors on the long time behavior of NLS on product spaces with a particular emphasis on the existence of solutions with growing higher Sobolev norms.

We study asymptotic behavior of radial solutions of a nonlocal Fokker-Planck equation describing the evolution of self-attracting particles. In particular, we consider stationary solutions in balls and in the whole space, self-similar solutions defined globally in time, blowing up self-similar solutions, and singularities of solutions that blow up in a finite time.

Radially symmetric solutions of a nonlocal Fokker-Planck equation describing the evolution of self-attracting particles in a bounded container are studied. Conditions ensuring either global-in-time existence of solutions or their finite time blow up are given.

We consider a biharmonic problem ${\Delta }^2{u}_{\omega }=f_{\omega }$ with Navier type boundary conditions $u_{\omega }=\Delta u_{\omega }=0$, on a family of truncated sectors ${\Omega }_{\omega }$ in ${ℝ}^2$ of radius $r$, $0<r<1$ and opening angle $\omega$, $\omega \in (2\pi /3,\pi ]$ when $\omega$ is close to $\pi$. The family of right-hand sides ${\left(f_{\omega }\right)}_{\omega \in (2\pi /3,\pi ]}$ is assumed to depend smoothly on $\omega$ in $L^2\left({\Omega }_{\omega }\right)$. The main result is that $u_{\omega }$ converges to $u_{\pi }$ when $\omega \to \pi$ with respect to the $H^2$-norm. We can also show that the $H^2$-topology is optimal for such a convergence result.

An elliptic system in ${ℝ}^n$, which is invariant under the action of the group ${\mathbb{Z}}^n$ is considered. We construct a holomorphic family of finite-dimensional subrepresentations of the group in the space of solutions (Floquet solutions), such that any solution of the growth $O\left(\exp \right(a\left|x\right|\left)\right)$ at infinity can be rewritten in the form of an integral over the family.

Let G be a Lie group. The main new result of this paper is an estimate in L2 (G) for the Davies perturbation of the semigroup generated by a centered sublaplacian H on G. When G is amenable, such estimates hold only for sublaplacians which are centered. Our semigroup estimate enables us to give new proofs of Gaussian heat kernel estimates established by Varopoulos on amenable Lie groups and by Alexopoulos on Lie groups of polynomial growth.

Let G be a Lie group of polynomial volume growth. Consider a differential operator H of order 2m on G which is a sum of even powers of a generating list $A₁,...,A_{d^{\text{'}}}$ of right invariant vector fields. When G is solvable, we obtain an algebraic condition on the list $A₁,...,A_{d^{\text{'}}}$ which is sufficient to ensure that the semigroup kernel of H satisfies global Gaussian estimates for all times. For G not necessarily solvable, we state an analytic condition on the list which is necessary and sufficient for global Gaussian estimates....

We propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses ε are at the order of the skin depth or essentially smaller. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t. to...

We propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses ε are at the order of the skin depth or essentially smaller. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t. to...

If ${\left(e^{-tA}\right)}_{t>0}$ is a strongly continuous and contractive semigroup on a complex Banach space $B$, then $-{(-A)}^{\alpha }$, $0<\alpha <1$, generates a holomorphic semigroup on $B$. This was proved by K. Yosida in [7]. Using similar techniques, we present a class $H$ of Bernstein functions such that for all $f\in H$, the operator $-f(-A)$ generates a holomorphic semigroup.