Matthias Hofmann

• Texas A&M University

We investigate the existence or non-existence of spectral minimal partitions of unbounded metric graphs, where the operator applied to each of the partition elements is a Schrödinger operator of the form -\Delta + V with suitable (electric) potential V , which is taken as a fixed, underlying function on the whole graph. We show that there is a stro...

We investigate the existence or non-existence of spectral minimal partitions of unbounded metric graphs, where the operator applied to each of the partition elements is a Schr\"odinger operator of the form $-\Delta + V$ with suitable (electric) potential $V$, which is taken as a fixed, underlying ``landscape'' on the whole graph. We show that there...

We prove interlacing inequalities between spectral minimal energies of metric graphs built on Dirichlet and standard Laplacian eigenvalues, as recently introduced in Kennedy et al. (Calc Var PDE 60:61, 2021). These inequalities, which involve the first Betti number and the number of degree one vertices of the graph, recall both interlacing and othe...

We establish metric graph counterparts of Pleijel’s theorem on the asymptotics of the number of nodal domains $$\nu _n$$ ν n of the n th eigenfunction(s) of a broad class of operators on compact metric graphs, including Schrödinger operators with $$L^1$$ L 1 -potentials and a variety of vertex conditions as well as the p -Laplacian with natural ver...

We study properties of spectral minimal partitions of metric graphs within the framework recently introduced in Kennedy et al. (Calc Var 60:6, 2021). We provide sharp lower and upper estimates for minimal partition energies in different classes of partitions; while the lower bounds are reminiscent of the classic isoperimetric inequalities for metri...

We prove interlacing inequalities between spectral minimal energies of metric graphs built on Dirichlet and standard Laplacian eigenvalues, as recently introduced in [Kennedy et al, arXiv:2005.01126]. These inequalities, which involve the first Betti number and the number of degree one vertices of the graph, recall both interlacing and other inequa...

We establish metric graph counterparts of Pleijel's theorem on the asymptotics of the number of nodal domains $\nu_n$ of the $n$-th eigenfunction(s) of a broad class of operators on compact metric graphs, including Schr\"odinger operators with $L^1$-potentials and a variety of vertex conditions as well as the $p$-Laplacian with natural vertex condi...

We study properties of spectral minimal partitions of metric graphs within the framework recently introduced in [Kennedy et al. (2020), arXiv:2005.01126]. We provide sharp lower and upper estimates for minimal partition energies in different classes of partitions; while the lower bounds are reminiscent of the classic isoperimetric inequalities for...

The purpose of this paper is to develop a general existence theory for constrained minimization problems for functionals defined on function spaces on metric measure spaces $(\mathcal M, d, \mu)$. We apply this theory to functionals defined on metric graphs $\mathcal G$, in particular $L^2$-constrained minimization problems of the form $$E(u) = \fr...