Thomas Hogancamp

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Thesis Research

The focus of my research has been on applications of bifurcation theory to nonlinear partial differential equations. The equations I study have the following general structure: \(\nabla \cdot (\mathcal{W}'(|\nabla u|^{2})\nabla u)=0.\) In practice, one would specify suitable boundary conditions in addition to the PDE given above. The function $\mathcal{W}$ is interpreted as the strain energy density and $u$ as the displacement in the context of nonlinear elasticity theory (when studying gas dynamics, these function represent instead the density and velocity of a gas).

I am also interested in adapting tools from classical elliptic theory to degenerate elliptic PDEs. The Maximum Principle, the Hopf Lemma, and the standard machinery of elliptic regularity either fail outright or become vastly more complex in this case. Consequently, far less is known about the qualitative and quantitative properties of solutions to such problems. Insights from transonic gas dynamics suggest that stability and smoothness can’t be expected generally.

You can access a pre-print of my first published paper, Broadening Global Families of Anti-Plane Shear Equilibria, on the arXiv or through my Researchgate account. In this work, families of equilibrium solutions are constructed for several classes of materials. One class is shown to exhibit broadening. This is characterized by displacements which become increasingly wide and flat. Numerics predict related behavior for internal solitary water waves, but it remains a challenging open question to rigourously verifiy their existence. Another class of materials are shown to generate solutions that lose ellipticity; a phenomena that is that is intimately linked to shock formation.

My most recent research helps shed light on the consequences of degenerate ellipticity in the governing equations of anti-plane shear. Singularities are shown to develop as the strain reaches a critical state and ellipticity is lost. Degenerate elliptic PDEs, and more generally mixed elliptic-hyperbolic equations, continue to present serious mathematical challenges for many physically relevant problems.

An overview of some of my other mathematical interests can be found under the Other Projects/Interests tab.

Publications

Conference Presentations/Talks