This work explores free boundary characteristics in the no-sign obstacle problem, suggesting new mathematical methods.
We present several results inspired by the method recently developed by Figalli and Serra regarding the fine structure of the free boundary, applied here to the no-sign obstacle problem. After overcoming various technical challenges, this approach proves partially effective in addressing the no-sign obstacle problem. However, we also identify key questions that remain open and appear to require techniques not currently available to us.
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Jeon et al. (2026) studied this question.
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