Kazdan–Warner equation on graph
By Alexander Grigorʼyan, Yong Lin, Y. Yang. 2016
170 citations recorded by the source corpus.
Let $$G=(V,E)$$G=(V,E) be a connected finite graph and $$\Delta $$Δ be the usual graph Laplacian. Using the calculus of variations and a method of upper and lower solutions, we give various conditions such that the Kazdan–Warner equation $$\Delta u=c-he^u$$Δu=c-heu has a solution on V, where c is a constant, and $$h:V\rightarrow \mathbb {R}$$h:V→R is a function. We also consider similar equations involving higher order derivatives on graph. Our results can be compared with the original manifold case of Kazdan and Warner (Ann. Math. 99(1):14–47, 1974).
View Paper: https://www.semanticscholar.org/paper/18cf2e0c1639a16ba0411c64a429252e537af679
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Citations in the corpus
- Kazdan-Warner equation on infinite graphs (2017, result): “ Kazdan and Warner [15] also discussed the solvability of (1.1) on certain non-compact two dimensional manifolds. The literature on the Kazdan-Warner equation is huge. However, in a very recent paper [13], Grigor’yan, Lin and Yang have studied this equation on a finite graph. In particular, in the re”
- Kazdan–Warner equation on graph in the negative case (2016, background): “Grigor’yan, Lin and Yang [5] showed that if the Kazdan-Warner equation has a solution, then h, the average value of h, is negative.”
- The Kazdan–Warner equation on canonically compactifiable graphs (2017, methodology): “The availability of these tools such as the Sobolev embedding theorem, a Poincaré inequality and a Trudinger–Moser inequality, which hold trivially in the finite setting, see [16], owes here to the assumption on the graph being canonically compactifiable and of finite measure.”