Nils Ackermann

A priori bounds, nodal equilibria and connecting orbits in indefinite superlinear parabolic problems

We consider the dynamics of the semiflow associated with a class of semilinear parabolic problems on a smooth bounded domain, posed with homogeneous Dirichlet boundary conditions. The distinguishing feature of this class is the indefinite superlinear (but subcritical) growth of the nonlinearity at infinity. We present new a priori bounds for global semiorbits that enable us to give dynamical proofs of known and new existence results for equilibria. In addition, we can prove the existence of connecting orbits in many cases. One advantage of our approach is that the parabolic semiflow is naturally order preserving, in contrast to pseudo-gradient flows considered when using variational methods. Therefore we can obtain much information on nodal properties of equilibria that was not known before.

Alternating Sign Multibump Solutions of Nonlinear Elliptic Equations in Expanding Tubular Domains

Let Γ denote a smooth simple curve in ℝ N , N ≥ 2, possibly with boundary. Let Ω R be the open normal tubular neighborhood of radius 1 of the expanded curve RΓ: = {Rx | x ∈ Γ∖∂Γ}. Consider the superlinear problem − Δu + λu = f(u) on the domains Ω R , as R → ∞, with homogeneous Dirichlet boundary condition. We prove the existence of multibump solutions with bumps lined up along RΓ with alternating signs. The function f is superlinear at 0 and at ∞, but it is not assumed to be odd. If the boundary of the curve is nonempty our results give examples of contractible domains in which the problem has multiple sign changing solutions.

A Cauchy-Schwarz type inequality for bilinear integrals on positive measures

If W: R n → [0, oo] is Borel measurable, define for σ-finite positive Borel measures μ, ν on R n the bilinear integral expression I(W; μ, ν):= ∫ R n∫ R W(x - y) dμ(x) dv(y) We give conditions on W such that there is a constant C > 0, independent of μ and ν, with I(W; μ, v) ≤ CI(W;μ, μ)I(W; v, ν). Our results apply to a much larger class of functions W than known before.