The Boundary Behavior of Heat Kernels of Dirichlet Laplacians

By Qi S. Zhang. 2002

138 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e4efe5c34de2e298588598c4e97ef24ded0f5157

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Critical SQG in Bounded Domains (2016, methodology): “Z∞ 0 f(x)−et∆f(x) t−1−s 2dt (29) for f∈ D((−ΛD)s). We use precise upper and lower bounds for the kernel HD(t,x,y)of the heat operator, (et∆f)(x) = Z Ω HD(t,x,y)f(y)dy. (30) These are as follows ([16],[24],[25]). There exists a time T>0 depending on the domain Ω and constants c, C, k, K, depending”
Heat kernel estimates for the fractional Laplacian with Dirichlet conditions (2009, background): “Explicit qualitatively sharp estimates for the classical heat kernel in C1,1 domains were established in 2002 by Zhang [42].”
Heat kernel estimates for the Dirichlet fractional Laplacian (2009, methodology): “(iii) In [27], a scale invariant parabolic boundary Harnack inequality obtained in [16] is used to obtain a sharp lower bound heat kernel estimate for the Dirichlet Laplacian.”
Dirichlet heat kernel estimates for Δα/2 + Δβ/2 (2010, background): “See [17], [18], [19] for upper bound estimates and [31] for the lower bound estimates of the Dirichlet heat kernels in bounded C(1,1) domains.”
Nonhomogeneous boundary conditions for the spectral fractional Laplacian (2015, unknown): “Then, GsΩ(x, z) δ(z) = 1 Γ(s) δ(z) ∫ ∞ 0 pΩ(t, x, z)t s−1 dt = |z − x|2s Γ(s) δ(z) ∫ ∞ 0 pΩ(|z − x| 2τ, x, z)τs−1 dτ Since Ω has C1,1 boundary, given x ∈ Ω, pΩ(·, x, ·) ∈ C1((0,+∞) × Ω) and the following heat kernel bound holds (cf. Davies, Simon and Zhang [9, 10, 24]): [ δ(x)δ(y) t ∧ 1 ] 1 c1tN/2 e”
Two-sided heat kernel estimates for censored stable-like processes (2010, background): “See [11–13] for upper bound estimates and [21] for the lower bound estimates of the Dirichlet heat kernels in bounded C1,1 domains.”
A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains (2013, background): “Davies and Simon [28, 30] (for the case s = 1), while the improved lower bound follows from the sharp lower heat kernel bounds of [46], see also [11] or Section 3 for more details.”
Potential Theory of Subordinate Brownian Motion (2009, background): “The lower bound was obtained in [62] and [54] and states that for any A > 0, there exist positive constants c2 and c such that for any t ∈ (0, A] and any x, y ∈ D, p(t, x, y) ≥ c2 ( ρ(x)ρ(y) t ∧ 1 ) t−d/2 exp ( −c|x− y| 2 t ) .”
Dirichlet heat kernel for unimodal Lévy processes (2014, methodology): “We recall that precise estimates for the heat kernel of the Laplacian (and the Brownian motion) were given for C1,1 domains in 2002 by Zhang [49].”