<p>We start the investigation of free boundary variational models featuring varying singularities. The theory depends strongly on the nature of the singular power <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\gamma (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and how it changes. Under a mild continuity assumption on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\gamma (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gamma (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> varies in a <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$W^{1,n^{+}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <msup> <mi>n</mi> <mo>+</mo> </msup> </mrow> </msup> </math></EquationSource> </InlineEquation> fashion, then the free boundary is locally a <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C^{1,\delta }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>δ</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> surface, up to a negligible singular set of Hausdorff co-dimension at least 3.</p>

On free boundary problems shaped by varying singularities

期刊论文
发表日期：

2025年12月6日

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Damião J. Araújo,
Aelson Sobral,
Eduardo V. Teixeira,
José Miguel Urbano

摘要

We start the investigation of free boundary variational models featuring varying singularities. The theory depends strongly on the nature of the singular power $\gamma (x)$ $γ (x)$ and how it changes. Under a mild continuity assumption on $\gamma (x)$ $γ (x)$ , 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 $\gamma (x)$ $γ (x)$ varies in a $W^{1,n^{+}}$ $W^{1, n^{+}}$ fashion, then the free boundary is locally a $C^{1,\delta }$ $C^{1, δ}$ surface, up to a negligible singular set of Hausdorff co-dimension at least 3.