Communications in Partial Differential Equations· 2026Q1
An elliptic-parabolic free boundary problem with discontinuous data
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- 2026year
Short summary
Discontinuous initial and boundary data in a 1+1D elliptic-parabolic free boundary problem lead to Lipschitz regularity in space and C1/2 regularity in time for weak solutions in the unsaturated region, with the free boundary also being C1/2.
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Key points
- Investigates the impact of discontinuous data on regularity in a 1+1D elliptic-parabolic free boundary problem.
- Weak solutions are Lipschitz in space and C1/2 in time in the unsaturated region when data is only bounded.
- The free boundary is shown to be locally the graph of a C1/2 function.
- The established C1/2 regularity for the free boundary is demonstrated to be optimal.
AI-generated from the title and abstract; the full text is not read.
Abstract
We consider an elliptic-parabolic free boundary problem that models the fluid flow through a partially saturated porous medium. The free boundary arises as the interface separating the saturated and unsaturated regions. Our main goal is to investigate, for the 1 + 1 dimensional model, how jump discontinuities on the boundary and initial data influence the regularity of both the solution and the free boundary. We show that if the data is merely bounded, then weak solutions are Lipschitz in space and C1/2 in time in the unsaturated region. Moreover, the free boundary is locally the graph of a C1/2 function, and this regularity is optimal. We view this analysis as a stepping stone toward the study of local regularization for higher-dimensional elliptic-parabolic free boundaries.
The authors' abstract, as published at the source. Communications in Partial Differential Equations, 2026 · DOI ↗
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Field: Applied Mathematics
Applied MathematicsMathematics