Remarks on the quadratic Hessian equation
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909617567563776 |
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| author | Mooney, Connor |
| author_facet | Mooney, Connor |
| contents | We prove that viscosity solutions to the quadratic Hessian equation $$σ_2(D^2u) = 1$$ cannot touch a harmonic function on a minimal surface from below. This can be viewed as a form of strict $2$-convexity. We also prove an a priori interior $C^2$ estimate in terms of the $W^{2,\,p}$ norm, for any $p > 2$. Finally, we discuss how these results rule out certain strategies for constructing counterexamples to regularity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_14586 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Remarks on the quadratic Hessian equation Mooney, Connor Analysis of PDEs 35J60, 35B65 We prove that viscosity solutions to the quadratic Hessian equation $$σ_2(D^2u) = 1$$ cannot touch a harmonic function on a minimal surface from below. This can be viewed as a form of strict $2$-convexity. We also prove an a priori interior $C^2$ estimate in terms of the $W^{2,\,p}$ norm, for any $p > 2$. Finally, we discuss how these results rule out certain strategies for constructing counterexamples to regularity. |
| title | Remarks on the quadratic Hessian equation |
| topic | Analysis of PDEs 35J60, 35B65 |
| url | https://arxiv.org/abs/2505.14586 |