Heat kernel estimates for the Dirichlet fractional Laplacian

Zhen-Qing Chen; Panki Kim; Renming Song

Displaying similar documents to “Heat kernel estimates for the Dirichlet fractional Laplacian”

Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball

Vladimir Varlamov (2000)

Studia Mathematica

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The nonlinear heat equation with a fractional Laplacian $[u_{t} + {(- Δ)}^{α / 2} u = u^{2}, 0 < α \leq 2]$ , is considered in a unit ball $B$ . Homogeneous boundary conditions and small initial conditions are examined. For 3/2 + ε₁ ≤ α ≤ 2, where ε₁ > 0 is small, the global-in-time mild solution from the space $C ⁰ ([0, \infty), H ₀^{κ} (B))$ with κ < α - 1/2 is constructed in the form of an eigenfunction expansion series. The uniqueness is proved for 0 < κ < α - 1/2, and the higher-order long-time asymptotics is calculated.

Inequalities involving heat potentials and Green functions

Neil A. Watson (2015)

Mathematica Bohemica

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We take some well-known inequalities for Green functions relative to Laplace’s equation, and prove not only analogues of them relative to the heat equation, but generalizations of those analogues to the heat potentials of nonnegative measures on an arbitrary open set $E$ whose supports are compact polar subsets of $E$ . We then use the special case where the measure associated to the potential has point support, in the following situation. Given a nonnegative supertemperature on an open set...

Fractional Laplacian with singular drift

Tomasz Jakubowski (2011)

Studia Mathematica

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For α ∈ (1,2) we consider the equation $\partial_{t} u = Δ^{α / 2} u + b \cdot \nabla u$ , where b is a time-independent, divergence-free singular vector field of the Morrey class $M ₁^{1 - α}$ . We show that if the Morrey norm ${| | b | |}_{M ₁^{1 - α}}$ is sufficiently small, then the fundamental solution is globally in time comparable with the density of the isotropic stable process.

Gradient estimates of Li Yau type for a general heat equation on Riemannian manifolds

Nguyen Ngoc Khanh (2016)

Archivum Mathematicum

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In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds $(M, g)$ for the following general heat equation $u_{t} = Δ_{V} u + a u log u + b u$ where $a$ is a constant and $b$ is a differentiable function defined on $M \times [0, \infty)$ . We suppose that the Bakry-Émery curvature and the $N$ -dimensional Bakry-Émery curvature are bounded from below, respectively. Then we obtain the gradient estimate of Li-Yau type for the above general heat equation. Our results generalize the work of Huang-Ma ([4]) and Y. Li ([6]), recently. ...

Fractional integral operators on $B^{p, λ}$ with Morrey-Campanato norms

Katsuo Matsuoka, Eiichi Nakai (2011)

Banach Center Publications

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We introduce function spaces $B^{p, λ}$ with Morrey-Campanato norms, which unify $B^{p, λ}$ , $C M O^{p, λ}$ and Morrey-Campanato spaces, and prove the boundedness of the fractional integral operator $I_{α}$ on these spaces.

Explicit solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity

María F. Natale, Domingo A. Tarzia (2006)

Bollettino dell'Unione Matematica Italiana

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We study a one-phase Stefan problem for a semi-infinite material with temperature-dependent thermal conductivity with a constant temperature or a heat flux condition of the type $-q_{0}/\sqrt{t}$ ( $q_{0}>0$ ) at the fixed face $x=0$ . We obtain in both cases sufficient conditions for data in order to have a parametric representation of the solution of the similarity type for $t\geq t_{0}>0$ with $t_{0}$ an arbitrary positive time. These explicit solutions are obtained through the unique solution of an integral equation with the time...

Initial measures for the stochastic heat equation

Daniel Conus, Mathew Joseph, Davar Khoshnevisan, Shang-Yuan Shiu (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider a family of nonlinear stochastic heat equations of the form $\partial_{t} u = ℒ u + σ (u) \dot{W}$ , where $\dot{W}$ denotes space–time white noise, $ℒ$ the generator of a symmetric Lévy process on $𝐑$ , and $σ$ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure $u_{0}$ . Tight a priori bounds on the moments of the solution are also obtained. In the particular case that $ℒ f = c f^{''}$ for some $c g t; 0$ , we prove that if $u_{0}$ is a finite measure of compact support, then the...

Hydrodynamical behavior of symmetric exclusion with slow bonds

Tertuliano Franco, Patrícia Gonçalves, Adriana Neumann (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the exclusion process in the one-dimensional discrete torus with $N$ points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance $N^{- β}$ , with $β \in [0, \infty)$ . We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter $β$ . If $β \in [0, 1)$ , the hydrodynamic limit is given by the usual heat equation. If $β = 1$ , it is given by a parabolic equation involving an operator...

Stochastic differential equations with Sobolev drifts and driven by $α$ -stable processes

Xicheng Zhang (2013)

Annales de l'I.H.P. Probabilités et statistiques

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In this article we prove the pathwise uniqueness for stochastic differential equations in $ℝ^{d}$ with time-dependent Sobolev drifts, and driven by symmetric $α$ -stable processes provided that $α \in (1, 2)$ and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when $α \in (\frac{2 d}{d + 1}, 2)$ . Our proof is based on some estimates of Krylov’s type for purely discontinuous semimartingales.

Hardy's theorem for the helgason Fourier transform on noncompact rank one symmetric spaces

S. Thangavelu (2002)

Colloquium Mathematicae

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Let G be a semisimple Lie group with Iwasawa decomposition G = KAN. Let X = G/K be the associated symmetric space and assume that X is of rank one. Let M be the centraliser of A in K and consider an orthonormal basis $Y_{δ, j} : δ \in K ̂ ₀, 1 \leq j \leq d_{δ}$ of L²(K/M) consisting of K-finite functions of type δ on K/M. For a function f on X let f̃(λ,b), λ ∈ ℂ, be the Helgason Fourier transform. Let $h_{t}$ be the heat kernel associated to the Laplace-Beltrami operator and let $Q_{δ} (i λ + ϱ)$ be the Kostant polynomials. We establish the following...

$L^{p} - L^{q}$ boundedness of analytic families of fractional integrals

Valentina Casarino, Silvia Secco (2008)

Studia Mathematica

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We consider a double analytic family of fractional integrals $S_{z}^{γ, α}$ along the curve ${t \mapsto | t |}^{α}$ , introduced for α = 2 by L. Grafakos in 1993 and defined by $(S_{z}^{γ, α} f) (x ₁, x ₂) : = 1 / Γ (z + 1 / 2) {\int \int | u - 1 |}^{z} ψ (u - 1) {f (x ₁ - t, x ₂ - u | t |}^{α} {) d u | t |}^{γ} d t / t$ , where ψ is a bump function on ℝ supported near the origin, $f \in_{c}^{\infty} (ℝ ²)$ , z,γ ∈ ℂ, Re γ ≥ 0, α ∈ ℝ, α ≥ 2. We determine the set of all (1/p,1/q,Re z) such that $S_{z}^{γ, α}$ maps $L^{p} (ℝ ²)$ to $L^{q} (ℝ ²)$ boundedly. Our proof is based on product-type kernel arguments. More precisely, we prove that the kernel $K_{- 1 + i θ}^{i ϱ, α}$ is a product kernel on ℝ², adapted to the curve ${t \mapsto | t |}^{α}$ ; as a consequence, we show...

Property C for ODE and Applications to an Inverse Problem for a Heat Equation

A. G. Ramm (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let $ℓ_{j} : = - d ² / d x ² + k ² q_{j} (x)$ , k = const > 0, j = 1,2, $0 < e s s i n f q_{j} (x) \leq e s s s u p q_{j} (x) < \infty$ . Suppose that (*) $\int_{0}^{1} p (x) u ₁ (x, k) u ₂ (x, k) d x = 0$ for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0,1], which changes sign finitely many times, and $u_{j}$ solves the problem $ℓ_{j} u_{j} = 0$ , 0 ≤ x ≤ 1, $u_{j}^{'} (0, k) = 0$ , $u_{j} (0, k) = 1$ . It is proved that (*) implies p = 0. This result is applied to an inverse problem for a heat equation.

Nonanalyticity of solutions to $\partial_{t} u = \partial ²_{x} u + u ²$

Grzegorz Łysik (2003)

Colloquium Mathematicae

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It is proved that the solution to the initial value problem $\partial_{t} u = \partial ²_{x} u + u ²$ , u(0,x) = 1/(1+x²), does not belong to the Gevrey class $G^{s}$ in time for 0 ≤ s < 1. The proof is based on an estimation of a double sum of products of binomial coefficients.

A uniform dimension result for two-dimensional fractional multiplicative processes

Xiong Jin (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Given a two-dimensional fractional multiplicative process ${(F_{t})}_{t \in [0, 1]}$ determined by two Hurst exponents $H_{1}$ and $H_{2}$ , we show that there is an associated uniform Hausdorff dimension result for the images of subsets of $[0, 1]$ by $F$ if and only if $H_{1} = H_{2}$ .

Observability inequalities and measurable sets

Jone Apraiz, Luis Escauriaza, Gengsheng Wang, C. Zhang (2014)

Journal of the European Mathematical Society

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This paper presents two observability inequalities for the heat equation over $Ω \times (0, T)$ . In the first one, the observation is from a subset of positive measure in $Ω \times (0, T)$ , while in the second, the observation is from a subset of positive surface measure on $\partial Ω \times (0, T)$ . It also proves the Lebeau-Robbiano spectral inequality when $Ω$ is a bounded Lipschitz and locally star-shaped domain. Some applications for the above-mentioned observability inequalities are provided.

Generalized fractional integrals on central Morrey spaces and generalized λ-CMO spaces

Katsuo Matsuoka (2014)

Banach Center Publications

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We introduce the generalized fractional integrals $I ̃_{α, d}$ and prove the strong and weak boundedness of $I ̃_{α, d}$ on the central Morrey spaces $B^{p, λ} (ℝ ⁿ)$ . In order to show the boundedness, the generalized λ-central mean oscillation spaces $Λ_{p, λ}^{(d)} (ℝ ⁿ)$ and the generalized weak λ-central mean oscillation spaces $W Λ_{p, λ}^{(d)} (ℝ ⁿ)$ play an important role.

Classical boundary value problems for integrable temperatures in a $C^{1}$ domain

Anna Grimaldi Piro, Francesco Ragnedda (1991)

Annales Polonici Mathematici

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Abstract. We study a Neumann problem for the heat equation in a cylindrical domain with $C^{1}$ -base and data in $h_{c}^{1}$ , a subspace of $L$ 1. We derive our results, considering the action of an adjoint operator on $B_{T} M O C$ , a predual of $h_{c}^{1}$ , and using known properties of this last space.

Weighted estimates for the iterated commutators of multilinear maximal and fractional type operators

Qingying Xue (2013)

Studia Mathematica

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The following iterated commutators $T_{*, Π b}$ of the maximal operator for multilinear singular integral operators and $I_{α, Π b}$ of the multilinear fractional integral operator are introduced and studied: $T_{*, Π b} (f ⃗) (x) = s u p_{δ > 0} | [b ₁, [b ₂, \dots {[b_{m - 1}, [b ₘ, T_{δ}] ₘ]}_{m - 1} \dots] ₂] ₁ (f ⃗) (x) |$ , $I_{α, Π b} (f ⃗) (x) = [b ₁, [b ₂, \dots {[b_{m - 1}, [b ₘ, I_{α}] ₘ]}_{m - 1} \dots] ₂] ₁ (f ⃗) (x)$ , where $T_{δ}$ are the smooth truncations of the multilinear singular integral operators and $I_{α}$ is the multilinear fractional integral operator, $b_{i} \in B M O$ for i = 1,…,m and f⃗ = (f1,…,fm). Weighted strong and L(logL) type end-point estimates for the above iterated commutators associated with two classes of multiple...

Density of smooth maps for fractional Sobolev spaces $W^{s, p}$ into $ℓ$ simply connected manifolds when $s \geq 1$

Pierre Bousquet, Augusto C. Ponce, Jean Van Schaftingen (2013)

Confluentes Mathematici

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Given a compact manifold $N^{n} \subset ℝ^{ν}$ and real numbers $s \geq 1$ and $1 \leq p < \infty$ , we prove that the class $C^{\infty} ({\bar{Q}}^{m}; N^{n})$ of smooth maps on the cube with values into $N^{n}$ is strongly dense in the fractional Sobolev space $W^{s, p} (Q^{m}; N^{n})$ when $N^{n}$ is $⌊ s p ⌋$ simply connected. For $s p$ integer, we prove weak sequential density of $C^{\infty} ({\bar{Q}}^{m}; N^{n})$ when $N^{n}$ is $s p - 1$ simply connected. The proofs are based on the existence of a retraction of $ℝ^{ν}$ onto $N^{n}$ except for a small subset of $N^{n}$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s, p} \cap W^{1, s p}$ .

Fractional global domination in graphs

Subramanian Arumugam, Kalimuthu Karuppasamy, Ismail Sahul Hamid (2010)

Discussiones Mathematicae Graph Theory

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Let G = (V,E) be a graph. A function g:V → [0,1] is called a global dominating function (GDF) of G, if for every v ∈ V, $g (N [v]) = \sum_{u \in N [v]} g (u) \geq 1$ and $g (\bar{N (v)}) = \sum_{u \notin N (v)} g (u) \geq 1$ . A GDF g of a graph G is called minimal (MGDF) if for all functions f:V → [0,1] such that f ≤ g and f(v) ≠ g(v) for at least one v ∈ V, f is not a GDF. The fractional global domination number $γ_{f g} (G)$ is defined as follows: $γ_{f g} (G)$ = min|g|:g is an MGDF of G where $| g | = \sum_{v \in V} g (v)$ . In this paper we initiate a study of this parameter.

Second order elliptic operators with complex bounded measurable coefficients in $L^{p}$ , Sobolev and Hardy spaces

Steve Hofmann, Svitlana Mayboroda, Alan McIntosh (2011)

Annales scientifiques de l'École Normale Supérieure

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Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$ , such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in $L^{p}$ , Sobolev, and some new Hardy spaces naturally associated to $L$ . First, we show...