Browsing by Subject "Partial differential equations"

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    Boundary value problems for second-order elliptic equations and related topics.
    (2021-05) Poggi Cevallos, Bruno
    We study perturbation results for boundary value problems for second-order elliptic partial differential equations, and the exponential decay of solutions to generalized Schr\"odinger operators. First, through the use of sawtooth domains and the extrapolation technique of Carleson measures, we show the stability of the solvability of the Dirichlet problem for (additive) Carleson perturbations of certain degenerate elliptic operators $-\text{div }A\nabla$ on domains with low dimensional boundaries (joint work with S. Mayboroda). Then, with a different method of proof, we expand these perturbation results to more abstract domains (including some domains with mixed-dimensional boundaries) and a broader type of Carleson perturbation, yielding some new applications (including to free boundary problems) (joint work with J. Feneuil). Next, together with S. Bortz, S. Hofmann, J.L. Luna Garc\'ia, and S. Mayboroda, we consider the uniformly elliptic operators $L=-\text{div }(A\nabla+B_1)+B_2\nabla+V$ on the upper half space $\mathbb R^{n+1}_+=\mathbb R^n\times\{t>0\}$, $n\geq3$, with $t-$independent coefficients, and we prove the $L^2$ solvability of the Dirichlet, Neumann and Regularity problems under the condition that $|B_1|,|B_2|,|V|^2$ have small $L^n(\mathbb R^n)$ norm. Finally, we show that for generalized magnetic Schr\"odinger operators $-(\nabla-i{\bf a})^TA(\nabla-i{\bf a})+V$, with certain conditions providing an uncertainty principle, resolvents and Lax-Milgram solutions exhibit exponential decay (in an $L^2-$sense), and we improve these estimates to upper pointwise exponential decay for the magnetic Schr\"odinger operator $-(\nabla-i{\bf a})^2$, and to sharp (that is, upper and lower) pointwise exponential decay for the Schr\"odinger operator on a non-homogeneous medium $-\text{div }A\nabla +V$ (joint work with S. Mayboroda).
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    Two Problems Involving Random Walks on Graphs: Random surfers, PageRank, and short-time asymptotics for the heat kernel
    (2021-12) Yuan, Amber
    Semi-supervised and unsupervised machine learning methods often rely on graphs to model data, prompting research on how theoretical properties of operators on graphs are leveraged in learning problems. In the first part of the thesis, we propose a framework for rigorously studying continuum limits of learning algorithms on directed graphs. We use the new framework to study the PageRank and show how it can be interpreted as a numerical scheme on a directed graph involving a type of normalized graph Laplacian. We show that the corresponding continuum limit problem, which is taken as the number of webpages grows to infinity, is a second-order, elliptic equation that contains reaction, diffusion, and advection terms. In the second part of the thesis, we work in the undirected graph setting and study the short-term behavior of a graph-based random walk defined via the heat kernel. We prove how to estimate the random walk via a Gaussian and propose a method for homogenizing the graph Laplacian to obtain better length-scale restrictions for parameters in the graph model.