On free boundary problems shaped by varying singularities

TL;DR

This work develops a variational theory for free boundary problems with spatially varying singularities by studying minimizers of a non-differentiable energy with a variable exponent $\gamma(x)$. The authors establish existence and local $C^{1,\alpha}$-regularity of minimizers independent of the continuity of $\gamma$, derive sharp gradient and non-degeneracy bounds near the free boundary, and obtain optimal pointwise growth under weak continuity assumptions on $\gamma$. A Weiss-type monotonicity formula is introduced, allowing classification of blow-ups as $\beta(z_0)$-homogeneous and enabling a dimension-reduction argument that yields $C^{1,\delta}$ regularity of the free boundary away from a singular set of Hausdorff dimension at most $n-2$; stronger results hold when $\gamma$ and $\delta$ lie in $W^{1,n^+}$. The paper furthermore proves Hausdorff-measure estimates and porosity of the free boundary, underscoring the continuum of geometries arising from varying singularities and laying groundwork for analysis of heterogeneous media in applications.

Abstract

We start the investigation of free boundary variational models featuring varying singularities. The theory depends strongly on the nature of the singular power $γ(x)$ and how it changes. Under a mild continuity assumption on $γ(x)$, 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 $γ(x)$ varies in a $W^{1,n^{+}}$ fashion, then the free boundary is locally a $C^{1,δ}$ surface, up to a negligible singular set of Hausdorff co-dimension at least $2$.

Theorems & Definitions (53)

• Proposition 2.1
• proof
• Remark 2.1
• Theorem 2.1
• Lemma 2.1
• Lemma 2.2
• proof : Proof of Theorem \ref{['localregthm']}
• Remark 2.2
• Theorem 2.2
• proof