
Open access
Date
2023-02-15Type
- Working Paper
ETH Bibliography
yes
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Abstract
This paper revives a four-decade-old problem concerning regularity theory for (continuous) constraint maps with free boundaries. Dividing the map into two parts, the distance part and the projected image to the constraint, one can prove various properties for each component. As already pointed out in the literature, the distance part falls under the classical obstacle problem, which is well-studied by classical methods. A perplexing issue, untouched in the literature, is the properties of the projected image and its higher regularity, which we show to be at most of class $C^{2,1}$. In arbitrary dimensions, we prove that the image map is globally of class $W^{3,BMO}$, and locally of class $C^{2,1}$ around the regular part of the free boundary. The issue becomes more delicate around singular points, and we resolve it in two dimensions. In the appendix, we extend some of our results to what we call leaky maps. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000643671Publication status
publishedJournal / series
arXivPages / Article No.
Publisher
Cornell UniversityEdition / version
v1Subject
Analysis of PDEs (math.AP); Differential Geometry (math.DG); FOS: MathematicsOrganisational unit
09565 - Figalli, Alessio / Figalli, Alessio
Funding
721675 - Regularity and Stability in Partial Differential Equations (EC)
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Is previous version of: https://doi.org/10.3929/ethz-b-000693766
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ETH Bibliography
yes
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