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On Dirichlet-Schrödinger operators with strong potentials

Gabriele Grillo — 1995

Studia Mathematica

We consider Schrödinger operators H = -Δ/2 + V (V≥0 and locally bounded) with Dirichlet boundary conditions, on any open and connected subdomain D n which either is bounded or satisfies the condition d ( x , D c ) 0 as |x| → ∞. We prove exponential decay at the boundary of all the eigenfunctions of H whenever V diverges sufficiently fast at the boundary ∂D, in the sense that d ( x , D C ) 2 V ( x ) as d ( x , D C ) 0 . We also prove bounds from above and below for Tr(exp[-tH]), and in particular we give criterions for the finiteness of such trace....

On the Schrödinger heat kernel in horn-shaped domains

Gabriele Grillo — 2004

Colloquium Mathematicae

We prove pointwise lower bounds for the heat kernel of Schrödinger semigroups on Euclidean domains under Dirichlet boundary conditions. The bounds take into account non-Gaussian corrections for the kernel due to the geometry of the domain. The results are applied to prove a general lower bound for the Schrödinger heat kernel in horn-shaped domains without assuming intrinsic ultracontractivity for the free heat semigroup.

Super and ultracontractive bounds for doubly nonlinear evolution equations.

Matteo BonforteGabriele Grillo — 2006

Revista Matemática Iberoamericana

We use logarithmic Sobolev inequalities involving the p-energy functional recently derived in [15], [21] to prove L-L smoothing and decay properties, of supercontractive and ultracontractive type, for the semigroups associated to doubly nonlinear evolution equations of the form u = Δ(u) (with m(p - 1) ≥ 1) in an arbitrary euclidean domain, homogeneous Dirichlet boundary conditions being assumed. The bound are of the form ||u(t)|| ≤ C||u|| / t for any r ≤ q ∈ [1,+∞) and t >...

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