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Math Jaxed + minor corrections
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Daniele Tampieri
Daniele Tampieri
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Boundness Boundedness and convergence

If I know that \Phi_\varepsilon$\Phi_\varepsilon$ is bounded in L^{\infty}(\mathbb{R}^{2d})$L^{\infty}(\mathbb{R}^{2d})$ and that \nabla \Phi_\varepsilon$\nabla \Phi_\varepsilon$ is bounded in L^{\infty}(\mathbb{R}^{2d})$L^{\infty}(\mathbb{R}^{2d})$, is it true that \nabla \Phi_\varepsilon \to \nabla \Phi$\nabla \Phi_\varepsilon \to \nabla \Phi$, where \Phi$\Phi$ is the limit of \Phi_\varepsilon$\Phi_\varepsilon$ as $\varepsilon \to 0$ (in the weak sense)?

Boundness and convergence

If I know that \Phi_\varepsilon is bounded in L^{\infty}(\mathbb{R}^{2d}) and that \nabla \Phi_\varepsilon is bounded in L^{\infty}(\mathbb{R}^{2d}), is it true that \nabla \Phi_\varepsilon \to \nabla \Phi, where \Phi is the limit of \Phi_\varepsilon (in the weak sense)?

Boundedness and convergence

If I know that $\Phi_\varepsilon$ is bounded in $L^{\infty}(\mathbb{R}^{2d})$ and that $\nabla \Phi_\varepsilon$ is bounded in $L^{\infty}(\mathbb{R}^{2d})$, is it true that $\nabla \Phi_\varepsilon \to \nabla \Phi$, where $\Phi$ is the limit of $\Phi_\varepsilon$ as $\varepsilon \to 0$ (in the weak sense)?

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Markus
Markus
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Boundness and convergence

If I know that \Phi_\varepsilon is bounded in L^{\infty}(\mathbb{R}^{2d}) and that \nabla \Phi_\varepsilon is bounded in L^{\infty}(\mathbb{R}^{2d}), is it true that \nabla \Phi_\varepsilon \to \nabla \Phi, where \Phi is the limit of \Phi_\varepsilon (in the weak sense)?