Nonlinear evolution equation associated with hypergraph Laplacian

Let V$$ V $$ be a finite set, E subset of 2V$$ E\subset {2} circumflex V $$ be a set of hyperedges, and w:E ->(0,infinity)$$ w:E\to \left(0,\infty \right) $$ be an edge weight. On the (wighted) hypergraph G=(V,E,w)$$ G equal to \left(V,E,w\right) $$, we can define a multivalued nonlinear operator LG,p$$ {L}_{G,p} $$ (p is an element of[1,infinity)$$ p\in \left[1,\infty \right) $$) as the subdifferential of a convex function on Double-struck capital RV$$ {\mathbb{R}} circumflex V $$, 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 Poincare-Wirtinger inequality in PDEs. Next, we consider an ordinary differential equation on Double-struck capital RV$$ {\mathbb{R}} circumflex V $$ 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 Poincare-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.