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arXiv · 2401.08071

On free boundary problems shaped by varying singularities

Damião Araújo·Aelson Sobral·Eduardo V. Teixeira·José Miguel Urbano
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Abstract

We start the investigation of free boundary variational models featuring varying singularities. The theory depends strongly on the nature of the singular power $\gamma(x)$ and how it changes. Under a mild continuity assumption on $\gamma(x)$, we prove the optimal regularity of minimizers. Such estimates vary point-by-point, leading to a continuum of free boundary geometries. We also conduct an extensive analysis of the free boundary shaped by the singularities. Utilizing a new monotonicity formula, we show that if the singular power $\gamma(x)$ varies in a $W^{1,n^{+}}$ fashion, then the free boundary is locally a $C^{1,\delta}$ surface, up to a negligible singular set of Hausdorff co-dimension at least $2$.

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Damião Araújo, Aelson Sobral, Eduardo V. Teixeira, José Miguel Urbano. 2024-01-16. On free boundary problems shaped by varying singularities. https://arxiv.org/abs/2401.08071

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