Contour Limits and a “Gliding Hump” Argument
We investigate the behavior of solutions of second-order elliptic Dirichlet problems for a convex domain by using a “gliding hump” technique and prove that there are no contour limits at a specified point of the boundary of the domain. Then we consider two-dimensional domains which have a reentrant...
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MDPI AG
2025-05-01
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| author | Ammar Khanfer Kirk Eugene Lancaster |
| author_facet | Ammar Khanfer Kirk Eugene Lancaster |
| author_sort | Ammar Khanfer |
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| description | We investigate the behavior of solutions of second-order elliptic Dirichlet problems for a convex domain by using a “gliding hump” technique and prove that there are no contour limits at a specified point of the boundary of the domain. Then we consider two-dimensional domains which have a reentrant (i.e., nonconvex) corner at a point <i>P</i> of the boundary of the domain. Assuming certain comparison functions exist, we prove that for any solution of an appropriate Dirichlet problem on the domain whose graph has finite area, there are infinitely many curves of finite length in the domain ending at <i>P</i> along which the solution has a limit at <i>P</i>. We then prove that such behavior occurs for quasilinear operations with positive genre. |
| format | Article |
| id | doaj-art-9c9eff98e3c74a25a8e737c7a04c779c |
| institution | Matheson Library |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-9c9eff98e3c74a25a8e737c7a04c779c2025-06-25T13:28:15ZengMDPI AGAxioms2075-16802025-05-0114642510.3390/axioms14060425Contour Limits and a “Gliding Hump” ArgumentAmmar Khanfer0Kirk Eugene Lancaster1Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi ArabiaIndependent Researcher, Wichita, KS 67226, USAWe investigate the behavior of solutions of second-order elliptic Dirichlet problems for a convex domain by using a “gliding hump” technique and prove that there are no contour limits at a specified point of the boundary of the domain. Then we consider two-dimensional domains which have a reentrant (i.e., nonconvex) corner at a point <i>P</i> of the boundary of the domain. Assuming certain comparison functions exist, we prove that for any solution of an appropriate Dirichlet problem on the domain whose graph has finite area, there are infinitely many curves of finite length in the domain ending at <i>P</i> along which the solution has a limit at <i>P</i>. We then prove that such behavior occurs for quasilinear operations with positive genre.https://www.mdpi.com/2075-1680/14/6/425Dirichlet problemgliding humpradial limitsasymptotic valuescontour limits |
| spellingShingle | Ammar Khanfer Kirk Eugene Lancaster Contour Limits and a “Gliding Hump” Argument Axioms Dirichlet problem gliding hump radial limits asymptotic values contour limits |
| title | Contour Limits and a “Gliding Hump” Argument |
| title_full | Contour Limits and a “Gliding Hump” Argument |
| title_fullStr | Contour Limits and a “Gliding Hump” Argument |
| title_full_unstemmed | Contour Limits and a “Gliding Hump” Argument |
| title_short | Contour Limits and a “Gliding Hump” Argument |
| title_sort | contour limits and a gliding hump argument |
| topic | Dirichlet problem gliding hump radial limits asymptotic values contour limits |
| url | https://www.mdpi.com/2075-1680/14/6/425 |
| work_keys_str_mv | AT ammarkhanfer contourlimitsandaglidinghumpargument AT kirkeugenelancaster contourlimitsandaglidinghumpargument |