## Heat kernels on metric-measure spaces and an application to semilinear elliptic equations (2003)

Venue: | Trans. Amer. Math. Soc |

Citations: | 12 - 4 self |

### BibTeX

@ARTICLE{Hu03heatkernels,

author = {Jiaxin Hu and Ka-sing Lau},

title = {Heat kernels on metric-measure spaces and an application to semilinear elliptic equations},

journal = {Trans. Amer. Math. Soc},

year = {2003},

pages = {2065--2095}

}

### OpenURL

### Abstract

Abstract. We consider a metric measure space (M, d,µ) andaheat kernel pt(x, y) on M satisfying certain upper and lower estimates, which depend on two parameters α and β. We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space (M, d, µ). Namely, α is the Hausdorff dimension of this space, whereas β, called the walk dimension, is determined via the properties of the family of Besov spaces W σ,2 on M. Moreover, the parameters α and β are related by the inequalities 2 ≤ β ≤ α +1. We prove also the embedding theorems for the space W β/2,2, and use them to obtain the existence results for weak solutions to semilinear elliptic equations on M of the form −Lu + f(x, u) =g(x), where L is the generator of the semigroup associated with pt. The framework in this paper is applicable for a large class of fractal domains, including the generalized Sierpiński carpet in Rn. 1.

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Citation Context ...e Hille–Yosida theorem dom(L) is dense in L 2 ;furthermore,L is a selfadjoint, non-positive-definite operator, which follows from the fact that Tt is selfadjoint and contractive (see for example [8], =-=[12]-=-, [16], or [29, Theorem 1, p. 237]). 4.2. The Dirichlet form. For any t>0, define the quadratic form Et on L 2 by ( ) u − Ttu (4.4) Et [u] := ,u , t where ( , ) is the inner product in L 2 . An easy c... |

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Citation Context ...by the Hille–Yosida theorem dom(L) is dense in L 2 ;furthermore,L is a selfadjoint, non-positive-definite operator, which follows from the fact that Tt is selfadjoint and contractive (see for example =-=[8]-=-, [12], [16], or [29, Theorem 1, p. 237]). 4.2. The Dirichlet form. For any t>0, define the quadratic form Et on L 2 by ( ) u − Ttu (4.4) Et [u] := ,u , t where ( , ) is the inner product in L 2 . An ... |

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Citation Context ...rite f(s) ≃ g(s) if there is a constant c such that for all s ∈ S, c −1 g(s) ≤ f(s) ≤ cg(s).2070 ALEXANDER GRIGOR’YAN, JIAXIN HU, AND KA-SING LAU 2. Some examples Let l ≥ 3 be an integer and let M0 ==-=[0, 1]-=- n (n ≥ 2). We divide M0 into l n equal subcubes. Remove a symmetric pattern of subcubes from M0, and denote by M1 what remains. Repeat the same procedure for each subcube in M1: divide each subcube i... |

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Citation Context ...)dµ(y)dµ(x). In terms of the spectral resolution {Eλ} of the operator −L, Et can be expressed as follows: ∫ ∞ 1 − e Et [u] = 0 −tλ d‖Eλu‖ t 2 2, which implies that Et [u] is decreasing in t (see also =-=[7]-=-).2076 ALEXANDER GRIGOR’YAN, JIAXIN HU, AND KA-SING LAU Let us define a quadratic form E by (4.6) E[u] := lim t→0+ Et [u] (where the limit may be +∞ since E [u] ≥Et [u]) and its domain D (E) by D(E) ... |

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Citation Context ...we will assume that a heat kernel is defined on a metric measure space, and show that this implies many interesting consequences for analysis on such a space. A similar approach was used by M. Barlow =-=[3]-=- and K. Pietruska-Pa̷luba [23], although in their works a heat kernel was assumed to be the transition density of a diffusion process on M, and in [23] the underlying space M was a subset of R n . Let... |

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Citation Context ...ence of sets {Mk}. Set and define ˜M = M = ∞⋂ k=0 Mk ∞⋃ l k M, ˜ k=0 wherewewriteaK = {ax : x ∈ K} for a real number a and a set K. The set M is called an unbounded generalized Sierpiński carpet (cf. =-=[5]-=-); see Figure 1, which corresponds to the case n =2andl =3. Figure 1. Generalized Sierpiński carpet The distance d on M is set to be the Euclidean distance, and the measure µ is the Hausdorff measure ... |

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Citation Context ...Φ1, Φ2 of the form ( Φ(s) =c exp −Cs β ) β−1 (see [5]). There are also plenty of other fractals such that (2.1) holds; see, for example, [3], [11].HEAT KERNELS ON METRIC MEASURE SPACES 2071 See also =-=[13]-=-, [14], [20] for the heat kernel estimates in the setting of graphs or manifolds. 3. Volume of balls Definition 3.1. We say that a heat kernel pt on a metric measure space (M,d,µ) satisfies the hypoth... |

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Citation Context ...a posteriori they happen to be the invariants of the space (M,d,µ) itself, provided the function Φ2 decays at ∞ sufficiently fast. The parameter α happens to be the Hausdorff dimension of M (see also =-=[10]-=-). The nature of the parameter β is more complicated. We will call it the walk dimension of the heat kernel pt. This terminology is motivated by the following observation: if the heat kernel pt is the... |

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Citation Context ... , which is a particular case of (1.5) with the functions Φ1, Φ2 of the form ( Φ(s) =c exp −Cs β ) β−1 (see [5]). There are also plenty of other fractals such that (2.1) holds; see, for example, [3], =-=[11]-=-.HEAT KERNELS ON METRIC MEASURE SPACES 2071 See also [13], [14], [20] for the heat kernel estimates in the setting of graphs or manifolds. 3. Volume of balls Definition 3.1. We say that a heat kernel... |

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Citation Context ...H1) canbe replaced by (H0). Furthermore, (iii) follows from (3.2) alone, in which case β may be any number larger than α. With this understanding, the statement of (iii) is not new (see, for example, =-=[6]-=-, [15], [21]) and is included for completeness. However, if the hypothesis (H1) does hold, then it follows from Theorem 4.2 that any function in D (E) isHölder continuous. Remark. If (M,d) satisfies t... |

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Citation Context ...s = ∞. On the other hand, the domain of the energy form associated with the operator − (−∆) β/2 is known to be another Besov space Lip (β/2, 2, 2), which is smaller than Lip (β/2, 2, ∞) =W β/2,2 (see =-=[2]-=-, [23], [25]). M. Barlow studied in [3] heat kernels on geodesic metric spaces, assuming that a heat kernel pt is the transition density of a diffusion process and that it satisfies (1.5) with functio... |

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Citation Context ...] and [14], using techniques that are not available for general metric spaces. Our contribution is that we prove (1.11) under the hypothesis (1.10), which seems to be nearly optimal. Barlow proved in =-=[4]-=- that for every pair α, β satisfying (1.11) there exists a random walk satisfying a discrete time version of (1.5) with these parameters. There is no doubt that the same is true for continuous time he... |