Nonlinear evolution equation associated with hypergraph Laplacian.

Let V$$ V $$ be a finite set, E⊂2V$$ E\subset {2}^V $$ be a set of hyperedges, and w:E→(0,∞)$$ w:E\to \left(0,\infty \right) $$ be an edge weight. On the (wighted) hypergraph G=(V,E,w)$$ G=\left(V,E,w\right) $$, we can define a multivalued nonlinear operator LG,p$$ {L}_{G,p} $$ (p∈[1,∞)$$ p\in \left[1,\infty \right) $$) as the subdifferential of a convex function on ℝV$$ {\mathbb{R}}^V $$, which is called "hypergraph p$$ p $$‐Laplacian." In this article, we first introduce an inequality for this operator LG,p$$ {L}_{G,p} $$, which resembles the Poincaré–Wirtinger inequality in PDEs. Next, we consider an ordinary differential equation on ℝV$$ {\mathbb{R}}^V $$ governed by LG,p$$ {L}_{G,p} $$, which is referred as "heat" equation on the graph and used to study the geometric structure of the hypergraph in recent researches. With the aid of the Poincaré–Wirtinger type inequality, we can discuss the existence and the large time behavior of solutions to the ODE by procedures similar to those for the standard heat equation in PDEs with the zero Neumann boundary condition. [ABSTRACT FROM AUTHOR]