Graduation Year
2016
Document Type
Thesis
Degree
Ph.D.
Degree Name
Doctor of Philosophy (Ph.D.)
Degree Granting Department
Mathematics and Statistics
Major Professor
D. Khavinson, Ph.D.
Committee Member
Catherine A. Bénéteau, Ph.D.
Committee Member
Arthur Danielyan, Ph.D.
Committee Member
Sherwin Kouchekian, Ph.D.
Committee Member
David Rabson, Ph.D.
Keywords
Eigenfunctions, Eigenvalues, Isoperimetric inequality, Plemelj-Sokhotski theorem, Single layer operator, Schatten ideals, Singular numbers.
Abstract
We show that the singular numbers of single layer potentials on smooth curves asymptotically behave like O(1/n). For the curves with singularities, as long as they contain a smooth sub-arc, the resulting single layer potentials are never trace-class. We provide upper bounds for the operator and the Hilbert-Schmidt norms of single layer potentials on smooth and chord-arc curves. Regarding the injectivity of single layer potentials on planar curves, we prove that among single layer potentials on dilations of a given curve, only one yields a non-injective single layer potential. A criterion for injectivity of single layer potentials on ellipses is given. We establish an isoperimetric inequality for Schatten p−norms of logarithmic potentials over quadrilaterals and its analogue for Newtonian potentials on parallelepipeds.
Scholar Commons Citation
Zoalroshd, Seyed, "On Spectral Properties of Single Layer Potentials" (2016). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/6445
