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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 $D \subset ℝ^{n}$ which either is bounded or satisfies the condition $d (x, D^{c}) \to 0$ 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 $d {(x, D^{C})}^{2} V (x) \to \infty$ as $d (x, D^{C}) \to 0$ . 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.

Simulated annealing with time-dependent energy function.

Alberto Frigerio; Gabriele Grillo — 1993

Mathematische Zeitschrift

Super and ultracontractive bounds for doubly nonlinear evolution equations.

Matteo Bonforte; Gabriele 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 >...

Direct and Reverse Gagliardo-Nirenberg Inequalities from Logarithmic Sobolev Inequalities

Matteo Bonforte; Gabriele Grillo — 2005

Bulletin of the Polish Academy of Sciences. Mathematics

We investigate the connection between certain logarithmic Sobolev inequalities and generalizations of Gagliardo-Nirenberg inequalities. A similar connection holds between reverse logarithmic Sobolev inequalities and a new class of reverse Gagliardo-Nirenberg inequalities.

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