Global Schauder estimates for a class of degenerate Kolmogorov equations

Enrico Priola

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Displaying similar documents to “Global Schauder estimates for a class of degenerate Kolmogorov equations”

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We study existence, uniqueness, and smoothing properties of the solutions to a class of linear second order elliptic and parabolic differential equations with unbounded coefficients in $ℝ^{n}$ . The main results are global Schauder estimates, which hold in spite of the unboundedness of the coefficients.

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We prove the existence of global attractors for the following semilinear degenerate parabolic equation on $ℝ^{N}$ : ∂u/∂t - div(σ(x)∇ u) + λu + f(x,u) = g(x), under a new condition concerning the variable nonnegative diffusivity σ(·) and for an arbitrary polynomial growth order of the nonlinearity f. To overcome some difficulties caused by the lack of compactness of the embeddings, these results are proved by combining the tail estimates method and the asymptotic a priori estimate method. ...

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We investigate the $L_{p}$ -spectrum of linear operators defined consistently on $L_{p} (Ω)$ for p₀ ≤ p ≤ p₁, where (Ω,μ) is an arbitrary σ-finite measure space and 1 ≤ p₀ < p₁ ≤ ∞. We prove p-independence of the $L_{p}$ -spectrum assuming weighted norm estimates. The assumptions are formulated in terms of a measurable semi-metric d on (Ω,μ); the balls with respect to this semi-metric are required to satisfy a subexponential volume growth condition. We show how previous results on $L_{p}$ -spectral independence...

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Hörmander’s famous Fourier multiplier theorem ensures the $L_{p}$ -boundedness of $F (- Δ_{ℝ} D)$ whenever $F \in ℋ (s)$ for some $s > \frac{D}{2}$ , where we denote by $ℋ (s)$ the set of functions satisfying the Hörmander condition for $s$ derivatives. Spectral multiplier theorems are extensions of this result to more general operators $A \geq 0$ and yield the $L_{p}$ -boundedness of $F (A)$ provided $F \in ℋ (s)$ for some $s$ sufficiently large. The harmonic oscillator $A = - Δ_{ℝ} + x^{2}$ shows that in general $s > \frac{D}{2}$ is not sufficient even if $A$ has a heat kernel satisfying gaussian estimates. In...

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We prove that for symbols in the modulation spaces $ℳ^{p, q}$ , p ≥ q, the associated multilinear pseudodifferential operators are bounded on products of appropriate modulation spaces. In particular, the symbols we study here are defined without any reference to smoothness, but rather in terms of their time-frequency behavior.

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Annales Polonici Mathematici

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In the recent years, many results have been established on positive solutions for boundary value problems of the form $- d i v (| \nabla u (x) |^{p - 2} \nabla u (x)) = λ f (u (x))$ in Ω, u(x)=0 on ∂Ω, where λ > 0, Ω is a bounded smooth domain and f(s) ≥ 0 for s ≥ 0. In this paper, a priori estimates of positive radial solutions are presented when N > p > 1, Ω is an N-ball or an annulus and f ∈ C¹(0,∞) ∪ C⁰([0,∞)) with f(0) < 0 (non-positone).

Compactness properties of Feller semigroups

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Fonctions biharmoniques adjointes

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The study of the equation (L₂L₁)*h = 0 or of the equivalent system L*₂h₂ = -h₁, L*₁h₁ = 0, where $L_{j} (j = 1, 2)$ is a second order elliptic differential operator, leads us to the following general framework: Starting from a biharmonic space, for example the space of solutions (u₁,u₂) of the system L₁u₁ = -u₂, L₂u₂ = 0, $L_{j} (j = 1, 2)$ being elliptic or parabolic, and by means of its Green pairs, we construct the associated adjoint biharmonic space which is in duality with the initial one.

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Johannes Sjöstrand (1976)

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Let $P$ be a classical pseudodifferential operator of order $m$ on a paracompact $C^{\infty}$ manifold $X$ . Let $p_{m}$ be the principal symbol and assume that $Σ = p_{m}^{- 1} (0)$ is an involutive $C^{\infty}$ sub-manifold of $T^{*} X ∖ 0$ , satisfying a certain transversality condition. We assume that $p_{m}$ vanishes exactly to order $M$ on $Σ$ and that the derivatives of order $M$ satisfy a certain condition, inspired from the Calderòn uniqueness theorem (usually empty when $M = 2$ ). Suppose that a Levi condition is valid for the lower order symbols. If $u \in 𝒟^{'} (X)$ , $P u \in C^{\infty} (X)$ , then...

Single-point blow-up for a semilinear parabolic system

Ph. Souplet (2009)

Journal of the European Mathematical Society

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We consider positive solutions of the system $u_{t} - Δ u = v^{p}$ ; $v_{t} - Δ v = u^{q}$ in a ball or in the whole space, with $p, q > 1$ . Relatively little is known on the blow-up set for semilinear parabolic systems and, up to now, no result was available for this basic system except for the very special case $p = q$ . Here we prove single-point blow-up for a large class of radial decreasing solutions. This in particular solves a problem left open in a paper of A. Friedman and Y. Giga (1987). We also obtain lower pointwise estimates for...