An Analog of the Neumann Problem for the 1-Laplace Equation in the Metric Setting: Existence, Boundary Regularity, and Stability
We study an inhomogeneous Neumann boundary value problem for functions of least gradient on bounded domains in metric spaces that are equipped with a doubling measure and support a Poincaré inequality. We show that solutions exist under certain regularity assumptions on the domain, but are generally...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2018-02-01
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| Series: | Analysis and Geometry in Metric Spaces |
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| Online Access: | https://doi.org/10.1515/agms-2018-0001 |
| _version_ | 1818716950113550336 |
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| author | Lahti Panu Malý Lukáš Shanmugalingam Nageswari |
| author_facet | Lahti Panu Malý Lukáš Shanmugalingam Nageswari |
| author_sort | Lahti Panu |
| collection | DOAJ |
| description | We study an inhomogeneous Neumann boundary value problem for functions of least gradient on bounded domains in metric spaces that are equipped with a doubling measure and support a Poincaré inequality. We show that solutions exist under certain regularity assumptions on the domain, but are generally nonunique. We also show that solutions can be taken to be differences of two characteristic functions, and that they are regular up to the boundary when the boundary is of positive mean curvature. By regular up to the boundary we mean that if the boundary data is 1 in a neighborhood of a point on the boundary of the domain, then the solution is −1 in the intersection of the domain with a possibly smaller neighborhood of that point. Finally, we consider the stability of solutions with respect to boundary data. |
| first_indexed | 2024-12-17T19:27:24Z |
| format | Article |
| id | doaj.art-e08e033471b34aa3b88160e01502ef4c |
| institution | Directory Open Access Journal |
| issn | 2299-3274 |
| language | English |
| last_indexed | 2024-12-17T19:27:24Z |
| publishDate | 2018-02-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Analysis and Geometry in Metric Spaces |
| spelling | doaj.art-e08e033471b34aa3b88160e01502ef4c2022-12-21T21:35:21ZengDe GruyterAnalysis and Geometry in Metric Spaces2299-32742018-02-016113110.1515/agms-2018-0001agms-2018-0001An Analog of the Neumann Problem for the 1-Laplace Equation in the Metric Setting: Existence, Boundary Regularity, and StabilityLahti Panu0Malý Lukáš1Shanmugalingam Nageswari2Department of Mathematical Sciences, P. O. Box 210025, University of Cincinnati, Cincinnati, OH 45221-0025, USADepartment of Mathematical Sciences, P. O. Box 210025, University of Cincinnati, Cincinnati, OH 45221-0025, USADepartment of Mathematical Sciences, P. O. Box 210025, University of Cincinnati, Cincinnati, OH 45221-0025, USAWe study an inhomogeneous Neumann boundary value problem for functions of least gradient on bounded domains in metric spaces that are equipped with a doubling measure and support a Poincaré inequality. We show that solutions exist under certain regularity assumptions on the domain, but are generally nonunique. We also show that solutions can be taken to be differences of two characteristic functions, and that they are regular up to the boundary when the boundary is of positive mean curvature. By regular up to the boundary we mean that if the boundary data is 1 in a neighborhood of a point on the boundary of the domain, then the solution is −1 in the intersection of the domain with a possibly smaller neighborhood of that point. Finally, we consider the stability of solutions with respect to boundary data.https://doi.org/10.1515/agms-2018-0001bounded variationmetric measure spaceneumann problempositive mean curvaturestability30l9926b3043a85 |
| spellingShingle | Lahti Panu Malý Lukáš Shanmugalingam Nageswari An Analog of the Neumann Problem for the 1-Laplace Equation in the Metric Setting: Existence, Boundary Regularity, and Stability Analysis and Geometry in Metric Spaces bounded variation metric measure space neumann problem positive mean curvature stability 30l99 26b30 43a85 |
| title | An Analog of the Neumann Problem for the 1-Laplace Equation in the Metric Setting: Existence, Boundary Regularity, and Stability |
| title_full | An Analog of the Neumann Problem for the 1-Laplace Equation in the Metric Setting: Existence, Boundary Regularity, and Stability |
| title_fullStr | An Analog of the Neumann Problem for the 1-Laplace Equation in the Metric Setting: Existence, Boundary Regularity, and Stability |
| title_full_unstemmed | An Analog of the Neumann Problem for the 1-Laplace Equation in the Metric Setting: Existence, Boundary Regularity, and Stability |
| title_short | An Analog of the Neumann Problem for the 1-Laplace Equation in the Metric Setting: Existence, Boundary Regularity, and Stability |
| title_sort | analog of the neumann problem for the 1 laplace equation in the metric setting existence boundary regularity and stability |
| topic | bounded variation metric measure space neumann problem positive mean curvature stability 30l99 26b30 43a85 |
| url | https://doi.org/10.1515/agms-2018-0001 |
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