A note on Hölder regularity of weak solutions to linear elliptic equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929336813092864 |
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| author | Adimurthi, Karthik |
| author_facet | Adimurthi, Karthik |
| contents | In this paper, we show that weak solutions of $$-\text{div} \mathbb{A}(x)\nabla u = 0 \qquad \text{where}\quad \mathbb{A}(x)= \mathbb{A}(x)^T \,\, \text{and} \,\, λ|ζ|^2 \leq \langle \mathbb{A}(x)ζ,ζ\rangle \leq Λ|ζ|^2,$$
and $\mathbb{A}(x) \equiv \mathbb{A}$ is a constant matrix are Hölder continuous $u \in C^α_{\text{loc}}$ with $α\geq \frac12 \left(-(n-2) + \sqrt{(n-2)^2 + \frac{4(n-1)λ}Λ} \right)$. This implies that the example constructed by Piccinini - Spagnolo is sharp in the class of constant matrices $\mathbb{A}(x) \equiv \mathbb{A}$. The proof of Hölder regularity does not go through a reduction of oscillation type argument and instead is achieved through a monotonicity formula.
In the case of general matrices $\mathbb{A}(x)$, we obtain the same regularity under some additional hypothesis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_03802 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on Hölder regularity of weak solutions to linear elliptic equations Adimurthi, Karthik Analysis of PDEs In this paper, we show that weak solutions of $$-\text{div} \mathbb{A}(x)\nabla u = 0 \qquad \text{where}\quad \mathbb{A}(x)= \mathbb{A}(x)^T \,\, \text{and} \,\, λ|ζ|^2 \leq \langle \mathbb{A}(x)ζ,ζ\rangle \leq Λ|ζ|^2,$$ and $\mathbb{A}(x) \equiv \mathbb{A}$ is a constant matrix are Hölder continuous $u \in C^α_{\text{loc}}$ with $α\geq \frac12 \left(-(n-2) + \sqrt{(n-2)^2 + \frac{4(n-1)λ}Λ} \right)$. This implies that the example constructed by Piccinini - Spagnolo is sharp in the class of constant matrices $\mathbb{A}(x) \equiv \mathbb{A}$. The proof of Hölder regularity does not go through a reduction of oscillation type argument and instead is achieved through a monotonicity formula. In the case of general matrices $\mathbb{A}(x)$, we obtain the same regularity under some additional hypothesis. |
| title | A note on Hölder regularity of weak solutions to linear elliptic equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.03802 |