Geodesic
On second-order and fourth-order elliptic systems consisting of bulk and surface PDEs: Well-posedness, regularity theory and eigenvalue problems
Patrik Knopf1 orcid ; Chun Liu2
1University of Regensburg, Germany
2Illinois Institute of Technology, Chicago, USA
Interfaces and free boundaries, Tome 23 (2021) no. 4, pp. 507-533

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DOI

In this paper, we study second-order and fourth-order elliptic problems which include not only a Poisson equation in the bulk but also an inhomogeneous Laplace–Beltrami equation on the boundary of the domain. The bulk and the surface PDE are coupled by a boundary condition that is either of Dirichlet or Robin type. We point out that both the Dirichlet and the Robin type boundary condition can be handled simultaneously through our formalism without having to change the framework. Moreover, we investigate the eigenvalue problems associated with these second-order and fourth-order elliptic systems.We further discuss the relation between these elliptic problems and certain parabolic problems, especially the Allen–Cahn equation and the Cahn–Hilliard equation with dynamic boundary conditions.
DOI : 10.4171/ifb/463
Classification : 35-XX, 58-XX
Mots-clés : Poisson equation, Laplace–Beltrami equation, bulk-surface coupling, Robin boundary condition, Dirichlet boundary condition, regularity theory, eigenvalue problem

Patrik Knopf  1   ; Chun Liu  2

1 University of Regensburg, Germany
2 Illinois Institute of Technology, Chicago, USA 
Patrik Knopf; Chun Liu. On second-order and fourth-order elliptic systems consisting of bulk and surface PDEs: Well-posedness, regularity theory and eigenvalue problems. Interfaces and free boundaries, Tome 23 (2021) no. 4, pp. 507-533. doi: 10.4171/ifb/463
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     author = {Patrik Knopf and Chun Liu},
     title = {On second-order and fourth-order elliptic systems consisting of bulk and surface {PDEs:} {Well-posedness,} regularity theory and eigenvalue problems},
     journal = {Interfaces and free boundaries},
     pages = {507--533},
     year = {2021},
     volume = {23},
     number = {4},
     doi = {10.4171/ifb/463},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/463/}
}
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