arXiv subjects
Publications and source records attributed to Ikemefuna Agbanusi.
We consider heat semigroups of the form $\exp(t(Δ- λ\mathbf{1}_{Ω_0}))$ on bounded domains. These singularly perturbed equations arise in certain models of diffusion limited chemical reactions. Using variants of Moser iteration, we show sub-exponential decay in the ``large coupling limit", i.e. as $λ\nearrow\infty$, in compact subdomains of the ``obstacle", $Ω_0$.
How large is the Bessel potential, $G_{α,μ}f$, compared to the Riesz potential, $I_αf$? In this paper, we show that if $I_αf\in L^p$ with $0<α<1$ and $p>1$, then the following interpolation bound holds: \[\Vert G_{α,μ}f\Vert_p\leq C(ω(I_αf,1/μ)_p)^α\cdot\Vert I_αf\Vert^{1-α}_p.\] Here $ω(f,t)_p$ is the $L^p$ modulus of continuity. However, if $α=p=1$, we obtain the ``$L\log L$" type result \[\Vert G_{1,μ}f\Vert_1\leq Bω(I_1f,1/μ)_1|\logω(I_1f,1/μ)_1|.\] These and other estimates are obtained by studying the quotient of the two operators, $E_{α,μ}:=\frac{(-Δ)^{α/2}}{(μ^2-Δ)^{α/2}}$. This operator is of independent interest due to its connection to approximation theory.
Banach famously related the smoothness of a function to the size of its level sets. More precisely, he showed that a continuous function is of bounded variation exactly when its "indicatrix" is integrable. In a similar vein, we connect the smoothness of the function -- measured now by its integral modulus of continuity -- to the structure of its superlevel sets. Our approach ultimately reduces to a continuum incidence problem for quantifying the regularity of open sets. The pay off is a refinement of Banach's original theorem and an answer to a question of Garsia--Sawyer.
This note investigates the regularity of zero-extensions of $L^p$ functions from bounded domains. Simple examples show the possibility of a loss in smoothness and our goal is to quantify this loss more generally. For the unit cube $\mathcal{Q}=[0,1]^d$, one of our main results is a bound for the $L^p$ modulus of continuity of zero-extensions. Using this, we prove that nonconstant functions in the Besov space $B^α_{p,q}(\mathcal{Q})$ have zero-extensions in $B^β_{p,r}(\mathbb{R}^d)$ with $β=\fracα{αp+1}$ and $r=q(1+αp)$. This seems to be new when $\frac1p\leqα<1$. The key idea behind the main estimate is to use piecewise constant approximation on dyadic subcubes. This technique can likely be sharpened, even for the unit cube, and extended to less regular domains.
A number of recent papers have considered signed graph Laplacians, a generalization of the classical graph Laplacian, where the edge weights are allowed to take either sign. In the classical case, where the edge weights are all positive, the Laplacian is positive semi-definite with the dimension of the kernel representing the number of connected components of the graph. In many applications one is interested in establishing conditions which guarantee the positive semi-definiteness of the matrix. In this paper we present an inequality on the eigenvalues of a weighted graph Laplacian (where the weights need not have any particular sign) in terms of the first two moments of the edge weights. This bound involves the eigenvalues of the equally weighted Laplacian on the graph as well as the eigenvalues of the adjacency matrix of the line graph (the edge-to-vertex dual graph). For a regular graph the bound can be expressed entirely in terms of the second eigenvalue of the equally weighted Laplacian, an object that has been extensively studied in connection with expander graphs and spectral measures of graph connectivity. We present several examples including Erdős-Rényi random graphs in the critical and subcritical regimes, random large $d$-regular graphs, and the complete graph, for which the inequalities here are tight.
We estimate the rate of convergence, in the so-called large coupling limit, for Schrödinger type operators on bounded domains. The Schrödinger we deal with have "interaction potentials" supported in a compact inclusion. We show that if the boundary of the inclusion is sufficiently smooth, one essentially recovers the "free Hamiltonian" in the exterior domain with Dirichlet boundary conditions. In addition, we obtain a convergence rate, in $L^2$, that is $\mathcal{O}(λ^{-\frac{1}{4}})$ where $λ$ is the coupling parameter. Our methods include energy estimates, trace estimates, interpolation and duality.
We propose a model for social dynamics on a network. In this model each actor holds a position on some issue, actors and their opinions being associated to vertices of the graph, and, additionally, the actors hold opinions of one another, with these opinions being associated to edges in the graph. These quantities are allowed to evolve according to the gradient flow of a natural free energy. We show that for a small spread in opinions the model converges to a consensus state, where all actors hold the same position. For a larger spread in opinion there is a phase transition marked by the birth of a second stable state: in addition to the consensus state there is a second polarized or partisan state. This state, when it exists, is conjectured to be global energy minimizer, with the consensus state being a local energy minimizer. We derive an energy inequality which supports this. Interestingly, all of the steady states we find, with the exception of the consensus state, are either balanced (in the sense of Heider) or are completely unbalanced states where all triangles are unbalanced. The latter solutions are, not surprisingly, always unstable.
In this paper we prove some new results and give new proofs of known results related to the large coupling limit for stationary Schrödinger operators. The operators we consider are of the form $-Δ+λV(x)$ where $Δ$ is the Laplacian, $V(x)$ is a real valued piecewise--constant potential having a jump discontinuity across a smooth interface and $λ$ is the coupling constant. Our main result is that the potential determines a non-local boundary condition on the interface and we systematically exploit this fact to derive various results about the large coupling problem. In particular, we obtain estimates for convergence rates and a description of the behavior of the spectrum of $-Δ+λV(x)$ as $λ\nearrow\infty$.