We consider Schrödinger operators H = -Δ/2 + V (V≥0 and locally bounded) with Dirichlet boundary conditions, on any open and connected subdomain $D\subset {\mathbb{R}}^{n}$ 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\left(x\right)\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....

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.

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 >...

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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