Mary Vaughan

Mary Vaughan

Ph.D.

Department of Mathematics
Texas State University


IMA Research Interests: Analysis and Partial Differential Equations

You can find all my papers and preprints on ArXiv.

Refereed papers
  1. A threshold for higher-order asymptotic development of genuinely nonlocal phase transition energies,
    to appear in Proc. Amer. Math. Soc. (with S. Dipierro and E. Valdinoci).
  2. The discrete dislocation dynamics of multiple dislocation loops,
    Arch. Ration. Mech. Anal. (2025) (with S. Patrizi).
  3. Interior Schauder estimates for fractional elliptic equations in nondivergence form,
    SIAM J. Math. Anal. (2025) (with P. R. Stinga).
  4. Continuous symmetrizations and uniqueness of solutions to nonlocal equations,
    Analysis & PDE (2025) (with M. G. Delgadino).
  5. A convergence result for the derivation of front propagation in nonlocal phase field models,
    Interfaces Free Bound. (2024) (with S. Patrizi).
  6. Fractional elliptic equations in nondivergence form: definition, applications and Harnack inequality,
    J. Math. Pures Appl. (2021) (with P. R. Stinga).
  7. The dual Kaczmarz algorithm,
    Acta Appl Math. (2020) (with A. Aboud, E. Curl, S. N. Harding, and E. S. Weber).
  8. One-sided fractional derivatives, fractional Laplacians, and weighted Sobolev spaces,
    Nonlinear Anal. (2020) (with P. R. Stinga).

Submitted papers
  1. Bond failure criteria in peridynamics: nonequivalence of critical stretch and critical energy density criteria,
    (with P. Seleson and P. R. Stinga).
  2. Weighted nonlocal area functionals without the triangle inequality,
    (with S. Dipierro and E. Valdinoci).
  3. A new graph-directed construction of nonlocal energies on the unit interval,
    (with A. Aboud and P. Alonso Ruiz).
  4. Asymptotic expansion of a nonlocal phase transition energy,
    (with S. Dipierro, S. Patrizi, and E. Valdinoci).

Ph.D. Thesis: Here is a link to my dissertation.


Video Presentations


Research Training:
In 2018, I was one of twenty students selected to participate in the NSF Mathematical Sciences Graduate Internship. I interned at Oak Ridge National Laboratory under the direction of Pablo Seleson.
When referring to the nth order derivative dnu(t)/dtn, L'Höpital asked Leibniz in his 1965 letter
"What if n is 1/2?" Leibniz replied "It will lead to a paradox. From this apparent paradox,
one day useful consequences will be drawn."