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- Title
Properties of the free boundaries for the obstacle problem of the porous medium equations.
- Authors
Kim, Sunghoon; Lee, Ki-Ahm; Park, Jinwan
- Abstract
In this paper, we study the existence and interior W 2 , p -regularity of the solution, and the regularity of the free boundary ∂ { u > ϕ } to the obstacle problem of the porous medium equation, u t = Δ u m ( m > 1 ) with the obstacle function ϕ. The penalization method is applied to have the existence and interior regularity. To deal with the interaction between two free boundaries ∂ { u > ϕ } and ∂ { u > 0 } , we consider two cases on the initial data which make the free boundary ∂ { u > ϕ } separate from the free boundary ∂ { u > 0 } . Then the problem is converted into the obstacle problem for a fully nonlinear operator. Hence, the C 1 -regularity of the free boundary ∂ { u > ϕ } is obtained by the regularity theory of a class of obstacle problems for the general fully nonlinear operator.
- Subjects
POROUS materials; NONLINEAR operators; EQUATIONS; NONLINEAR equations
- Publication
Advances in Calculus of Variations, 2024, Vol 17, Issue 1, p209
- ISSN
1864-8258
- Publication type
Article
- DOI
10.1515/acv-2021-0113