The prescribed mean curvature measure equation in non-parametric form
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910765221412864 |
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| author | Leonardi, Gian Paolo Comi, Giovanni Eugenio |
| author_facet | Leonardi, Gian Paolo Comi, Giovanni Eugenio |
| contents | We introduce a weak formulation of the non-parametric prescribed mean curvature equation with measure data and show the existence and several properties of $BV$ solutions under natural assumptions on the prescribed measure. Our approach does not rely on approximate or viscosity-type solutions. It requires combining various ingredients, including Anzellotti's pairing theory for divergence-measure fields and its recent developments, a refinement of Anzellotti-Giaquinta approximation, and convex duality theory. We also prove a Gamma-convergence result valid for suitable smooth approximations of the prescribed measure, and a maximum principle for continuous weak solutions. We finally construct some examples of non-uniqueness, showing at the same time the need for the continuity assumption in the maximum principle and an unexpected feature of weak solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_10592 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The prescribed mean curvature measure equation in non-parametric form Leonardi, Gian Paolo Comi, Giovanni Eugenio Analysis of PDEs 49Q10 (Primary) 35J93, 49Q20, 46N10 (Secondary) We introduce a weak formulation of the non-parametric prescribed mean curvature equation with measure data and show the existence and several properties of $BV$ solutions under natural assumptions on the prescribed measure. Our approach does not rely on approximate or viscosity-type solutions. It requires combining various ingredients, including Anzellotti's pairing theory for divergence-measure fields and its recent developments, a refinement of Anzellotti-Giaquinta approximation, and convex duality theory. We also prove a Gamma-convergence result valid for suitable smooth approximations of the prescribed measure, and a maximum principle for continuous weak solutions. We finally construct some examples of non-uniqueness, showing at the same time the need for the continuity assumption in the maximum principle and an unexpected feature of weak solutions. |
| title | The prescribed mean curvature measure equation in non-parametric form |
| topic | Analysis of PDEs 49Q10 (Primary) 35J93, 49Q20, 46N10 (Secondary) |
| url | https://arxiv.org/abs/2302.10592 |