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In this paper, we consider weak solutions of the Euler–Lagrange equation to a variational energy functional modeling the geometrically nonlinear Cosserat micropolar elasticity of continua in dimension three, which is a system coupling between the Poisson equation and the equation of $$p$$-harmonic maps (#2\le p\le 3$$). We show that if a weak solution is stationary, then its singular set is discrete for $2<3$ and has zero one-dimensional Hausdorff measure for $p=2$. If, in addition, it is a stable-stationary weak solution, then it is regular everywhere when $$p\in [2, 32/15]$$.
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Unique Equilibrium States for Geodesic Flows on Flat Surfaces with Singularities
Abstract Consider a compact surface of genus $$\geq 2$$ equipped with a metric that is flat everywhere except at finitely many cone points with angles greater than $$2\pi $$. Following the technique in the work of Burns, Climenhaga, Fisher, and Thompson, we prove that sufficiently regular potential functions have unique equilibrium states if the singular set does not support the full pressure. Moreover, we show that the pressure gap holds for any potential that is locally constant on a neighborhood of the singular set. Finally, we establish that the corresponding equilibrium states have the $$K$$-property and closed regular geodesics equidistribute.
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- Award ID(s):
- 1954463
- PAR ID:
- 10448463
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnac247
