Heat kernel bounds for the Laplacian on metric graphs of polygonal tilings
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Heat kernel bounds for the Laplacian on metric graphs of polygonal tilings René Pröpper
Received: 16 May 2012 / Accepted: 14 September 2012 / Published online: 25 September 2012 © Springer Science+Business Media New York 2012
Abstract We obtain an upper heat kernel bound for the Laplacian on metric graphs arising as one skeletons of certain polygonal tilings of the plane, which reflects the one dimensional as well as the two dimensional nature of these graphs. Keywords Heat kernels · Metric graphs · Tilings 1 Introduction If one looks at the heat semigroup on the metric graph with Kirchhoff type vertex conditions given by the uniform grid in R2 , i.e. the vertex set is Z2 and the edges are all straight-line segments in R2 between vertices (n, m) and (n , m ) ∈ Z2 with (n − n , m − m )2 = 1, then it is natural to guess that the semigroup behaves for small times like the one dimensional heat semigroup and for large times more like the two dimensional, i.e. the semigroup is governed by one and two dimensional Gaussian estimates. That this is indeed correct was shown in [10]. The behaviour of the heat semigroup on compact metric graphs with more general vertex conditions was treated in [9]. One would expect that the same result as for the uniform grid, which may be regarded as the boundary complex of the regular, edge-to-edge tiling of the plane with identical squares, is also true for the tiling with equilateral triangles or with regular hexagons or even more general tilings. Therefore, we have simplified the proof in [10] in some places and have thus enlarged the scope of the method a bit. The conditions we impose on the tilings are still rather restictive, but the tilings encompassed by our approach include some nice exemplars like triangulations of the Communicated by Jerome A. Goldstein. R. Pröpper () Institut für Analysis, Universität Ulm, Helmholtzstrasse 18, 89081 Ulm, Germany e-mail: [email protected]
Heat kernel bounds for the Laplacian on metric graphs
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plane, the 11 archimedean tilings, especially the three regular ones, and every other tiling composed from regular polygons of uniformly bounded area (from above and below); see [6] for an abundance of examples. We want to mention that there is another well known approach (see e.g. [11]) to getting upper as well as lower Gaussian estimates; this is via the volume doubling property and a global Poincaré inequality. In [8] this approach is shown to be feasible for large classes of metric graphs through results in [7] and especially in [1]. The method we use in this paper is less generally applicable, but is in comparison more direct and seems to have some interest in its own right. Now, we give a precise account of the setting we deal with. Let T = (Pn )n∈N be a tiling of the (euclidean) plane by convex, compact polygons Pn ⊂ R2 , n ∈ N, i.e. R2 = n∈N Pn and Pn ∩ Pm ⊂ ∂Pn for all n, m ∈ N with n = m, where ∂P denotes the boundary of the polygon P . We do not require the tiling is edge-to-edge (see [6]). The graph GT = n∈N ∂Pn of the po
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