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Lao
Lao
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How can I prove the following Liouville theorem for an elliptic equation with gradient perturbation?

Let $u \in L^2(\mathbb R^n;\mathbb R^n)$$u \in L^2(\mathbb R^n;\mathbb R)$ be a smooth solution of $$ -\Delta u + v \cdot \nabla u = 0 \qquad \text{in } \mathbb R^n$$ where we assume $\int_{\mathbb{R}^n}|\nabla v|^2 dx \leq C$ and $\mathrm{div}v=0$ for some $C > 0$, then $u$ is constant.

How can I prove the following Liouville theorem for an elliptic equation with gradient perturbation?

Let $u \in L^2(\mathbb R^n;\mathbb R^n)$ be a smooth solution of $$ -\Delta u + v \cdot \nabla u = 0 \qquad \text{in } \mathbb R^n$$ where we assume $\int_{\mathbb{R}^n}|\nabla v|^2 dx \leq C$ and $\mathrm{div}v=0$ for some $C > 0$, then $u$ is constant.

How can I prove the following Liouville theorem for an elliptic equation with gradient perturbation?

Let $u \in L^2(\mathbb R^n;\mathbb R)$ be a smooth solution of $$ -\Delta u + v \cdot \nabla u = 0 \qquad \text{in } \mathbb R^n$$ where we assume $\int_{\mathbb{R}^n}|\nabla v|^2 dx \leq C$ and $\mathrm{div}v=0$ for some $C > 0$, then $u$ is constant.

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Lao
Lao
  • 217
  • 1
  • 6

Liouville theorem for an elliptic equation with gradient perturbation

How can I prove the following Liouville theorem for an elliptic equation with gradient perturbation?

Let $u \in L^2(\mathbb R^n;\mathbb R^n)$ be a smooth solution of $$ -\Delta u + v \cdot \nabla u = 0 \qquad \text{in } \mathbb R^n$$ where we assume $\int_{\mathbb{R}^n}|\nabla v|^2 dx \leq C$ and $\mathrm{div}v=0$ for some $C > 0$, then $u$ is constant.