Quantitative Strong Maximum Principles for Fully Nonlinear Degenerate Elliptic Operators
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2025
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| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | This paper investigates the quantitative aspects of strong maximum principles for a class of fully nonlinear degenerate elliptic operators. While classical strong maximum principles assert that non-constant solutions cannot attain their maximum or minimum in the interior of the domain, quantitative versions provide explicit lower bounds on the modulus of continuity or growth estimates near the boundary or points of maximum. We focus on operators that may exhibit degeneracy, meaning their ellipticity can vanish at certain points or along specific directions, posing significant challenges to standard regularity theory and comparison arguments. Our analysis employs techniques from viscosity solutions and geometric measure theory, adapting them to handle the inherent non-linearity and degeneracy. We establish new quantitative strong maximum principles, providing specific estimates on the rate at which solutions must increase away from a minimum or decrease away from a maximum. These results are crucial for understanding the fine properties of solutions to a wide range of partial differential equations arising in areas such as optimal control, stochastic processes, and mathematical finance, where degeneracy is a common feature. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17806133 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Quantitative Strong Maximum Principles for Fully Nonlinear Degenerate Elliptic Operators Revista, Zen MATH, 10 This paper investigates the quantitative aspects of strong maximum principles for a class of fully nonlinear degenerate elliptic operators. While classical strong maximum principles assert that non-constant solutions cannot attain their maximum or minimum in the interior of the domain, quantitative versions provide explicit lower bounds on the modulus of continuity or growth estimates near the boundary or points of maximum. We focus on operators that may exhibit degeneracy, meaning their ellipticity can vanish at certain points or along specific directions, posing significant challenges to standard regularity theory and comparison arguments. Our analysis employs techniques from viscosity solutions and geometric measure theory, adapting them to handle the inherent non-linearity and degeneracy. We establish new quantitative strong maximum principles, providing specific estimates on the rate at which solutions must increase away from a minimum or decrease away from a maximum. These results are crucial for understanding the fine properties of solutions to a wide range of partial differential equations arising in areas such as optimal control, stochastic processes, and mathematical finance, where degeneracy is a common feature. |
| title | Quantitative Strong Maximum Principles for Fully Nonlinear Degenerate Elliptic Operators |
| url | https://doi.org/10.5281/zenodo.17806133 |