Quantitative unique continuation for real-valued solutions to second order elliptic equations in the plane
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916078230175744 |
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| author | Balc'h, Kévin Le Souza, Diego A. |
| author_facet | Balc'h, Kévin Le Souza, Diego A. |
| contents | In this article, we study a quantitative form of the Landis conjecture on exponential decay for real-valued solutions to second order elliptic equations with variable coefficients in the plane. In particular, we prove the following qualitative form of Landis conjecture, for $W_1, W_2 \in L^{\infty}(\mathbb R^2;\mathbb R^2)$, $V \in L^{\infty}(\mathbb R^2;\mathbb R)$ and $u \in H_{\mathrm{loc}}^{1}(\mathbb R^2)$ a real-valued weak solution to $-Δu - \nabla \cdot ( W_1 u ) +W_2 \cdot \nabla u + V u = 0$ in $\mathbb R^2$, satisfying for $δ>0$, $|u(x)| \leq \exp(- |x|^{1+δ})$, $x \in \mathbb R^2$, then $u \equiv 0$. Our methodology of proof is inspired by the one recently developed by Logunov, Malinnikova, Nadirashvili, and Nazarov that have treated the equation $-Δu + V u = 0$ in $\mathbb R^2$. Nevertheless, several differences and additional difficulties appear. New weak quantitative maximum principles are established for the construction of a positive multiplier in a suitable perforated domain, depending on the nodal set of $u$. The resulted divergence elliptic equation is then transformed into a non-homogeneous $\partial_{\overline{z}}$ equation thanks to a generalization of Stoilow factorization theorem obtained by the theory of quasiconformal mappings, an approximate type Poincaré lemma and the use of the Cauchy transform. Finally, a suitable Carleman estimate applied to the operator $\partial_{\overline{z}}$ is the last ingredient of our proof. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00441 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantitative unique continuation for real-valued solutions to second order elliptic equations in the plane Balc'h, Kévin Le Souza, Diego A. Analysis of PDEs Classical Analysis and ODEs Complex Variables 35B60, 35J15, 30C62 In this article, we study a quantitative form of the Landis conjecture on exponential decay for real-valued solutions to second order elliptic equations with variable coefficients in the plane. In particular, we prove the following qualitative form of Landis conjecture, for $W_1, W_2 \in L^{\infty}(\mathbb R^2;\mathbb R^2)$, $V \in L^{\infty}(\mathbb R^2;\mathbb R)$ and $u \in H_{\mathrm{loc}}^{1}(\mathbb R^2)$ a real-valued weak solution to $-Δu - \nabla \cdot ( W_1 u ) +W_2 \cdot \nabla u + V u = 0$ in $\mathbb R^2$, satisfying for $δ>0$, $|u(x)| \leq \exp(- |x|^{1+δ})$, $x \in \mathbb R^2$, then $u \equiv 0$. Our methodology of proof is inspired by the one recently developed by Logunov, Malinnikova, Nadirashvili, and Nazarov that have treated the equation $-Δu + V u = 0$ in $\mathbb R^2$. Nevertheless, several differences and additional difficulties appear. New weak quantitative maximum principles are established for the construction of a positive multiplier in a suitable perforated domain, depending on the nodal set of $u$. The resulted divergence elliptic equation is then transformed into a non-homogeneous $\partial_{\overline{z}}$ equation thanks to a generalization of Stoilow factorization theorem obtained by the theory of quasiconformal mappings, an approximate type Poincaré lemma and the use of the Cauchy transform. Finally, a suitable Carleman estimate applied to the operator $\partial_{\overline{z}}$ is the last ingredient of our proof. |
| title | Quantitative unique continuation for real-valued solutions to second order elliptic equations in the plane |
| topic | Analysis of PDEs Classical Analysis and ODEs Complex Variables 35B60, 35J15, 30C62 |
| url | https://arxiv.org/abs/2401.00441 |