Local Hölder Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912824283889664 |
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| author | Apaza, Juan Pablo Alcon |
| author_facet | Apaza, Juan Pablo Alcon |
| contents | We establish local boundedness and local Hölder continuity of weak solutions to the following prototype problem:
$$ -\operatorname{div}\left(|x|^{-2 β}|\nabla u|^{\mathbf{q}-2} \nabla u\right)+(-Δ)_{p(\cdot, \cdot), β}^{s(\cdot, \cdot)} u=0 \quad \text { in } \quad Ω, $$
where $Ω\subset \mathbb{R}^n, n \geq 2$, is a bounded domain. The nonlocal operator is defined by
$$ (-Δ)_{p(\cdot, \cdot), β}^{s(\cdot, \cdot)} u(x):=\mathrm{P} . \mathrm{V} . \int_Ω \frac{|u(x)-u(y)|^{p(x, y)-2}(u(x)-u(y))}{|x-y|^{n+s(x, y) p(x, y)}} \frac{1}{|x|^β|y|^β} \mathrm{d} y $$
Here, $p: Ω\times Ω\rightarrow(1, \infty)$ and $s: Ω\times Ω\rightarrow(0,1)$ are measurable functions, $\mathbf{q}:=\operatorname{ess}_{Ω\times Ω} p$, and $0 \leq β<n$. Our approach is analytic and relies on an adaptation of the De Giorgi-Nash-Moser theory to a mixed local-nonlocal framework with variable exponents and weights. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_01899 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local Hölder Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights Apaza, Juan Pablo Alcon Analysis of PDEs 35B45, 35B65, 35D30, 35J92, 35R11 We establish local boundedness and local Hölder continuity of weak solutions to the following prototype problem: $$ -\operatorname{div}\left(|x|^{-2 β}|\nabla u|^{\mathbf{q}-2} \nabla u\right)+(-Δ)_{p(\cdot, \cdot), β}^{s(\cdot, \cdot)} u=0 \quad \text { in } \quad Ω, $$ where $Ω\subset \mathbb{R}^n, n \geq 2$, is a bounded domain. The nonlocal operator is defined by $$ (-Δ)_{p(\cdot, \cdot), β}^{s(\cdot, \cdot)} u(x):=\mathrm{P} . \mathrm{V} . \int_Ω \frac{|u(x)-u(y)|^{p(x, y)-2}(u(x)-u(y))}{|x-y|^{n+s(x, y) p(x, y)}} \frac{1}{|x|^β|y|^β} \mathrm{d} y $$ Here, $p: Ω\times Ω\rightarrow(1, \infty)$ and $s: Ω\times Ω\rightarrow(0,1)$ are measurable functions, $\mathbf{q}:=\operatorname{ess}_{Ω\times Ω} p$, and $0 \leq β<n$. Our approach is analytic and relies on an adaptation of the De Giorgi-Nash-Moser theory to a mixed local-nonlocal framework with variable exponents and weights. |
| title | Local Hölder Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights |
| topic | Analysis of PDEs 35B45, 35B65, 35D30, 35J92, 35R11 |
| url | https://arxiv.org/abs/2507.01899 |