We show that a closed set <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {S}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">S</mi> </math></EquationSource> </InlineEquation> is removable for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Hölder continuous <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathscr {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>-harmonic functions in a domain <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> within a reversible Finsler manifold <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(\mathcal {M}, F, \texttt{V})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">M</mi> <mo>,</mo> <mi>F</mi> <mo>,</mo> <mi mathvariant="monospace">V</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of dimension <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. This removability holds under certain assumptions on <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$(\mathcal {M}, F, \texttt{V})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">M</mi> <mo>,</mo> <mi>F</mi> <mo>,</mo> <mi mathvariant="monospace">V</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the variable exponent p, provided that for every compact subset <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$K \subset \mathcal {S}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>⊂</mo> <mi mathvariant="script">S</mi> </mrow> </math></EquationSource> </InlineEquation>, the Hausdorff measure of K with dimension <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\text {n}_1 - p_K^+ + \alpha (p_K^+ - 1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>n</mtext> <mn>1</mn> </msub> <mo>-</mo> <msubsup> <mi>p</mi> <mi>K</mi> <mo>+</mo> </msubsup> <mo>+</mo> <mi>α</mi> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>p</mi> <mi>K</mi> <mo>+</mo> </msubsup> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is zero. Here, <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$p_K^+ = \sup _K p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>p</mi> <mi>K</mi> <mo>+</mo> </msubsup> <mo>=</mo> <msub> <mo movablelimits="true">sup</mo> <mi>K</mi> </msub> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\text {n}_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>n</mtext> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> is a dimension exponent satisfying the growth condition <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\texttt{V}(B(x, r)) \le \texttt{K}r^{\text {n}_1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="monospace">V</mi> <mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mi mathvariant="monospace">K</mi> <msup> <mi>r</mi> <msub> <mtext>n</mtext> <mn>1</mn> </msub> </msup> </mrow> </math></EquationSource> </InlineEquation> for all balls.The second main result establishes an estimate for <Equation ID="Equ100"> <EquationSource Format="TEX">$ \mu (B(x, r)) := \sup \left\{ \int _{B(x, r)} \mathscr {A}(\cdot , \varvec{\nabla }u) \bullet \mathscr {D}\zeta \; \text {d} \texttt{V}| 0 \le \zeta \le 1 \text{ and } \zeta \in C_0^{\infty }( B(x, r) ) \right\} , $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <mo movablelimits="true">sup</mo> <mfenced close="}" open="{"> <msub> <mo>∫</mo> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mrow> <mi mathvariant="script">A</mi> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo>,</mo> <mrow> <mi mathvariant="bold">∇</mi> </mrow> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>∙</mo> <mi mathvariant="script">D</mi> <mi>ζ</mi> <mspace width="0.277778em" /> <mtext>d</mtext> <mi mathvariant="monospace">V</mi> <mo stretchy="false">|</mo> <mn>0</mn> <mo>≤</mo> <mi>ζ</mi> <mo>≤</mo> <mn>1</mn> <mspace width="0.333333em" /> <mtext>and</mtext> <mspace width="0.333333em" /> <mi>ζ</mi> <mo>∈</mo> </mrow> <msubsup> <mi>C</mi> <mn>0</mn> <mi>∞</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </Equation>which is related to the measure <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\mu = \operatorname {div}(\mathscr {A} (\cdot , \varvec{\nabla }u))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>=</mo> <mo>div</mo> <mo stretchy="false">(</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo>,</mo> <mrow> <mi mathvariant="bold">∇</mi> </mrow> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

Removable sets for Hölder continuous solutions of $\mathcal {A}$-harmonic functions on Finsler manifolds

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2026年3月10日

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Juan Pablo Alcon Apaza

摘要

We show that a closed set $\mathcal {S}$ $S$ is removable for $\alpha $ $α$ -Hölder continuous $\mathscr {A}$ $A$ -harmonic functions in a domain $\Omega $ $Ω$ within a reversible Finsler manifold $(\mathcal {M}, F, \texttt{V})$ $(M, F, V)$ of dimension $n \ge 2$ $n \geq 2$ . This removability holds under certain assumptions on $(\mathcal {M}, F, \texttt{V})$ $(M, F, V)$ and the variable exponent p, provided that for every compact subset $K \subset \mathcal {S}$ $K \subset S$ , the Hausdorff measure of K with dimension $\text {n}_1 - p_K^+ + \alpha (p_K^+ - 1)$ $n_{1} - p_{K}^{+} + α (p_{K}^{+} - 1)$ is zero. Here, $p_K^+ = \sup _K p$ $p_{K}^{+} = {sup}_{K} p$ , and $\text {n}_1$ $n_{1}$ is a dimension exponent satisfying the growth condition $\texttt{V}(B(x, r)) \le \texttt{K}r^{\text {n}_1}$ $V (B (x, r)) \leq K r^{n_{1}}$ for all balls.

The second main result establishes an estimate for $ \mu (B(x, r)) := \sup \left\{ \int _{B(x, r)} \mathscr {A}(\cdot , \varvec{\nabla }u) \bullet \mathscr {D}\zeta \; \text {d} \texttt{V}| 0 \le \zeta \le 1 \text{ and } \zeta \in C_0^{\infty }( B(x, r) ) \right\} , $ $μ (B (x, r)) : = sup \{\int_{B (x, r)}, A (\cdot, \nabla u) ∙ D ζ d V | 0 \leq ζ \leq 1 and ζ \in, C_{0}^{\infty}, (B (x, r))\},$ which is related to the measure $\mu = \operatorname {div}(\mathscr {A} (\cdot , \varvec{\nabla }u))$ $μ = div (A (\cdot, \nabla u))$ .

Removable sets for Hölder continuous solutions of \(\mathcal {A}\)-harmonic functions on Finsler manifolds

摘要