heat kernel
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Author(s):  
Frederic Weber ◽  
Rico Zacher

AbstractWe establish a reduction principle to derive Li–Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. Our approach is not based on curvature-dimension inequalities but on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel. As an important application we solve a long-standing open problem by obtaining a Li–Yau inequality for positive solutions u to the fractional (in space) heat equation of the form $$(-\Delta )^{\beta /2}(\log u)\le C/t$$ ( - Δ ) β / 2 ( log u ) ≤ C / t , where $$\beta \in (0,2)$$ β ∈ ( 0 , 2 ) . We also show that this Li–Yau inequality allows to derive a Harnack inequality. We further illustrate our general result with an example in the discrete setting by proving a sharp Li–Yau inequality for diffusion on a complete graph.


Author(s):  
Diego Alonso-Orán ◽  
Fernando Chamizo ◽  
Ángel D. Martínez ◽  
Albert Mas

AbstractIn this paper we present an elementary proof of a pointwise radial monotonicity property of heat kernels that is shared by the Euclidean spaces, spheres and hyperbolic spaces. The main result was discovered by Cheeger and Yau in 1981 and rediscovered in special cases during the last few years. It deals with the monotonicity of the heat kernel from special points on revolution hypersurfaces. Our proof hinges on a non straightforward but elementary application of the parabolic maximum principle. As a consequence of the monotonicity property, we derive new inequalities involving classical special functions.


2021 ◽  
pp. 1-34
Author(s):  
G. Serafin

We establish short-time asymptotics with rates of convergence for the Laplace Dirichlet heat kernel in a ball. So far, such results were only known in simple cases where explicit formulae are available, i.e., for sets as half-line, interval and their products. Presented asymptotics may be considered as a complement or a generalization of the famous “principle of not feeling the boundary” in case of a ball. Following the metaphor, the principle reveals when the process does not feel the boundary, while we describe what happens when it starts feeling the boundary.


Author(s):  
Patrick Tolksdorf

AbstractWe investigate off-diagonal decay properties of the generalized Stokes semigroup with bounded measurable coefficients on $$\mathrm {L}^2_{\sigma } ({\mathbb {R}}^d)$$ L σ 2 ( R d ) . Such estimates are well-known for elliptic equations in the form of pointwise heat kernel bounds and for elliptic systems in the form of integrated off-diagonal estimates. On our way to unveil this off-diagonal behavior we prove resolvent estimates in Morrey spaces $$\mathrm {L}^{2 , \nu } ({\mathbb {R}}^d)$$ L 2 , ν ( R d ) with $$0 \le \nu < 2$$ 0 ≤ ν < 2 .


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