# The first eigenvalue of the $$p-$$ p - Laplacian on quantum graphs

The first eigenvalue of the $$p-$$ p - Laplacian on quantum graphs We study the first eigenvalue of the $$p-$$ p - Laplacian (with $$1<p<\infty$$ 1 < p < ∞ ) on a quantum graph with Dirichlet or Kirchoff boundary conditions on the nodes. We find lower and upper bounds for this eigenvalue when we prescribe the total sum of the lengths of the edges and the number of Dirichlet nodes of the graph. Also we find a formula for the shape derivative of the first eigenvalue (assuming that it is simple) when we perturb the graph by changing the length of an edge. Finally, we study in detail the limit cases $$p\rightarrow \infty$$ p → ∞ and $$p\rightarrow 1$$ p → 1 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# The first eigenvalue of the $$p-$$ p - Laplacian on quantum graphs

, Volume 6 (4) – Jan 29, 2016
27 pages

/lp/springer-journals/the-first-eigenvalue-of-the-p-p-laplacian-on-quantum-graphs-3laUKUvsif
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-016-0123-y
Publisher site
See Article on Publisher Site

### Abstract

We study the first eigenvalue of the $$p-$$ p - Laplacian (with $$1<p<\infty$$ 1 < p < ∞ ) on a quantum graph with Dirichlet or Kirchoff boundary conditions on the nodes. We find lower and upper bounds for this eigenvalue when we prescribe the total sum of the lengths of the edges and the number of Dirichlet nodes of the graph. Also we find a formula for the shape derivative of the first eigenvalue (assuming that it is simple) when we perturb the graph by changing the length of an edge. Finally, we study in detail the limit cases $$p\rightarrow \infty$$ p → ∞ and $$p\rightarrow 1$$ p → 1 .

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Jan 29, 2016