Laplace and bi-Laplace equations for directed networks and Markov chains
| Autor: | Thomas Hirschler, Wolfgang Woess |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Dirichlet problem
Pure mathematics Laplace transform General Mathematics 010102 general mathematics Boundary problem Probability (math.PR) Boundary (topology) Mathematics::Spectral Theory 01 natural sciences Mathematics - Analysis of PDEs Iterated function 31C20 35R02 60J10 Neumann boundary condition FOS: Mathematics 0101 mathematics Poisson's equation Laplace operator Mathematics - Probability Mathematics Analysis of PDEs (math.AP) |
| DOI: | 10.48550/arxiv.2104.01368 |
| Popis: | The networks of this -- primarily (but not exclusively) expository -- compendium are strongly connected, finite directed graphs $X$, where each oriented edge $(x,y)$ is equipped with a positive weight (conductance) $a(x,y)$. We are not assuming symmetry of this function, and in general we do not require that along with $(x,y)$, also $(y,x)$ is an edge. The weights give rise to a difference operator, the normalised version of which we consider as our Laplace operator. It is associated with a Markov chain with state space $X$. A non-empty subset of $X$ is designated as the boundary. We provide a systematic exposition of the different types of Laplace equations, starting with the Poisson equation, Dirichlet problem and Neumann problem. For the latter, we discuss the definition of outer normal derivatives. We then pass to Laplace equations involving potentials, thereby also addressing the Robin boundary problem. Next, we study the bi-Laplacian and associated equations: the iterated Poisson equation, the bi-Laplace Neumann and Dirichlet problems, and the "plate equation". It turns out that the bi-Laplace Dirichlet to Neumann map is of non-trivial interest. The exposition concludes with two detailed examples. Comment: Expositiones Mathematicae, to appear |
| Databáze: | OpenAIRE |
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