Graduation Year


Document Type




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.


Eigenfunctions, Eigenvalues, Isoperimetric inequality, Plemelj-Sokhotski theorem, Single layer operator, Schatten ideals, Singular numbers.


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.

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