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Connor Mooney
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Not necessarily- let let $\Omega = B_1 \cap \{x_3 > 0\}.$ Then $u(x) := (1-|x|^2)\frac{x_3}{|x|}$ is in $H^1_0(\Omega) \cap C^{\infty}(\Omega),$ but $u$ is discontinuous at the origin.

Not necessarily- let let $\Omega = B_1 \cap \{x_3 > 0\}.$ Then $u(x) := (1-|x|^2)\frac{x_3}{|x|}$ is in $H^1_0(\Omega) \cap C^{\infty}(\Omega),$ but $u$ is discontinuous at the origin.

Not necessarily- let $\Omega = B_1 \cap \{x_3 > 0\}.$ Then $u(x) := (1-|x|^2)\frac{x_3}{|x|}$ is in $H^1_0(\Omega) \cap C^{\infty}(\Omega),$ but $u$ is discontinuous at the origin.

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Connor Mooney
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Not necessarily- take any non-constant smooth function $w$ on $\mathbb{S}^2$ that vanishes on $\{x_3 = 0\}$, andlet let $\phi \in C^{\infty}_0(B_1)$ with$\Omega = B_1 \cap \{x_3 > 0\}.$ Then $\phi = 1$$u(x) := (1-|x|^2)\frac{x_3}{|x|}$ is in $B_{1/2}$. Then for $\Omega = B_1 \cap \{x_3 > 0\},$ we have $$u(x) := \phi(x)w(x/|x|) \in H^1_0(\Omega) \cap C^{\infty}(\Omega)$$$H^1_0(\Omega) \cap C^{\infty}(\Omega),$ but $u$ is discontinuous at the origin.

Not necessarily- take any non-constant smooth function $w$ on $\mathbb{S}^2$ that vanishes on $\{x_3 = 0\}$, and let $\phi \in C^{\infty}_0(B_1)$ with $\phi = 1$ in $B_{1/2}$. Then for $\Omega = B_1 \cap \{x_3 > 0\},$ we have $$u(x) := \phi(x)w(x/|x|) \in H^1_0(\Omega) \cap C^{\infty}(\Omega)$$ but $u$ is discontinuous at the origin.

Not necessarily- let let $\Omega = B_1 \cap \{x_3 > 0\}.$ Then $u(x) := (1-|x|^2)\frac{x_3}{|x|}$ is in $H^1_0(\Omega) \cap C^{\infty}(\Omega),$ but $u$ is discontinuous at the origin.

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Connor Mooney
  • 4.9k
  • 19
  • 16

Not necessarily- take any non-constant smooth function $w$ on $\mathbb{S}^2$ that vanishes on $\{x_3 = 0\}$, and let $\phi \in C^{\infty}_0(B_1)$ with $\phi = 1$ in $B_{1/2}$. Then for $\Omega = B_1 \cap \{x_3 > 0\},$ we have $$u(x) := \phi(x)w(x/|x|) \in H^1_0(\Omega) \cap C^{\infty}(\Omega)$$ but $u$ is discontinuous at the origin.