Two models forsandpile growth in weighted graphs
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914702626390016 |
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| author | Mazon, J. M. Toledo, J. |
| author_facet | Mazon, J. M. Toledo, J. |
| contents | In this paper we study $\infty$-Laplacian type diffusion equations in weighted graphs obtained as limit as $p\to \infty$ to two types of $p$-Laplacian evolution equations in such graphs. We propose these diffusion equations, that are governed by the subdifferential of a convex energy functionals associated to the indicator function of the set $$K^G_{\infty}:= \left\{ u \in L^2(V, ν_G) \ : \ \vert u(y) - u(x) \vert \leq 1 \ \ \hbox{if} \ \ x \sim y \right\}$$ and the set $$K^w_{\infty}:= \left\{ u \in L^2(V, ν_G) \ : \ \vert u(y) - u(x) \vert \leq \sqrt{w_{xy}} \ \ \hbox{if} \ \ x \sim y \right\}$$ as models for sandpile growth in weighted graphs. Moreover, we also analyse the collapse of the initial condition when it does not belong to the stable sets $K^G_{\infty}$ or $K^w_{\infty}$ by means of an abstract result given in~\cite{BEG}. We give an interpretation of the limit problems in terms of Monge-Kantorovich mass transport theory. Finally, we give some explicit solutions of simple examples that illustrate the dynamics of the sandpile growing or collapsing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02900 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two models forsandpile growth in weighted graphs Mazon, J. M. Toledo, J. Analysis of PDEs In this paper we study $\infty$-Laplacian type diffusion equations in weighted graphs obtained as limit as $p\to \infty$ to two types of $p$-Laplacian evolution equations in such graphs. We propose these diffusion equations, that are governed by the subdifferential of a convex energy functionals associated to the indicator function of the set $$K^G_{\infty}:= \left\{ u \in L^2(V, ν_G) \ : \ \vert u(y) - u(x) \vert \leq 1 \ \ \hbox{if} \ \ x \sim y \right\}$$ and the set $$K^w_{\infty}:= \left\{ u \in L^2(V, ν_G) \ : \ \vert u(y) - u(x) \vert \leq \sqrt{w_{xy}} \ \ \hbox{if} \ \ x \sim y \right\}$$ as models for sandpile growth in weighted graphs. Moreover, we also analyse the collapse of the initial condition when it does not belong to the stable sets $K^G_{\infty}$ or $K^w_{\infty}$ by means of an abstract result given in~\cite{BEG}. We give an interpretation of the limit problems in terms of Monge-Kantorovich mass transport theory. Finally, we give some explicit solutions of simple examples that illustrate the dynamics of the sandpile growing or collapsing. |
| title | Two models forsandpile growth in weighted graphs |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.02900 |