关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Real and Complex Analysis

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Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

Progress in Mathematics

On horizontal Hardy, Rellich, Caffarelli–Kohn–Nirenberg and p-sub-Laplacian inequalities on stratified groups

We prove some existence, nonexistence and regularity results for the boundary value problem Δ λ u + f ( u ) = 0 in  Ω , u ∣ ∂ Ω = 0 , where Ω is a bounded subset of R N , N ≥ 2 , and Δ λ is a Δ λ -Laplacian, i.e. a “degenerate” elliptic operator of the kind Δ λ : = ∑ i = 1 N ∂ x i ( λ i 2 ( x ) ∂ x i ) , λ = ( λ 1 , … , λ N ) . Together with some assumptions made in Franchi and Lanconelli (1984) [1] , the family λ is supposed to verify a condition making Δ λ homogeneous of degree two with respect to a group of dilations in R N .

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On semilinear Δλ-Laplace equation

We show an invariant Harnack inequality for a class of hypoelliptic
ultraparabolic operators with underlying homogeneous Lie group structures.
As a byproduct we prove a Liouville type theorem for the related “stationary”
operators. We also introduce a notion of link of homogeneous Lie Groups that
allows us to show that our results apply to wide classes of operators.

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An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations

We prove a Liouville-type theorem for a class of degenerate elliptic operators of the form L u ≔ ∑ i , j = 1 N ∂ x i ( a i j ∂ x j u ) + ∑ i = 1 N b i ∂ x i u . L is supposed to be X -elliptic, with respect to a family X = ( X 1 , … , X m ) of locally Lipschitz continuous vector fields, in the sense introduced in [E. Lanconelli, A.E. Kogoj, X -elliptic operators and X -control distances, Contributions in Honor of the Memory of Ennio De Giorgi, Ricerche di Matematica 49 (Suppl.) (2000) 223–243].

http://www.bcamath.org/documentos_public/archivos/publicaciones/liouvX1.pdf

Liouville theorem for X-elliptic operators

We consider degenerate parabolic equations of the form 

$$\left. \begin{array}{ll}\,\,\, \partial_t u = \Delta_\lambda u + f(u) \\u|_{\partial\Omega} = 0, u|_{t=0} = u_0\end{array}\right.$$

in a bounded domain ${\Omega\subset\mathbb{R}^N}$, where Δλ is a subelliptic operator of the type 

$$\quad \Delta_\lambda:= \sum_{i=1}^{N} \partial_{x_i}(\lambda_{i}^{2} \partial_{x_i}),\qquad \lambda = (\lambda_1,\ldots, \lambda_N).$$

We prove global existence of solutions and characterize their longtime behavior. In particular, we show the existence and finite fractal dimension of the global attractor of the generated semigroup and the convergence of solutions to an equilibrium solution when time tends to infinity.

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Attractors for a class of semi-linear degenerate parabolic equations

We derive Hardy-type inequalities for a large class of sub-elliptic operators that belong to the class of -Laplacians and find explicit values for the constants involved. Our results generalize previous inequalities obtained for Grushin-type operatorswhich were proved to be sharp.

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Alessia E. Kogoj

Papers

On semilinear Δλ-Laplace equation

An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations

Liouville theorem for X-elliptic operators

Attractors for a class of semi-linear degenerate parabolic equations

Hardy type inequalities for Δλ-Laplacians