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Davide Giraudo
Davide Giraudo
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density Density in fractional Sobolev space

Suppose $s∈(0,1)$, $D$ is an open set in $\mathbb{R}^d$. Define $$ H^s=(1−\Delta)^{-s/2}L^2(\mathbb{R}^d), $$$$ H^s=(1−\Delta)^{-s/2}L^2\left(\mathbb{R}^d\right), $$ $$ H^s_D=\{f\in H^s:f=0 \ a.e.\ on\ D^c\}. $$$$ H^s_D=\left\{f\in H^s:f=0 \mbox{ a.e. on } D^c\right\}. $$ Q: Is $C^\infty_c(D)$ dense in $H^s_D$(with norm $\|\cdot\|_{H^s}$) for any open set $D$?

Is there any element reference?

density in fractional Sobolev space

Suppose $s∈(0,1)$, $D$ is an open set in $\mathbb{R}^d$. Define $$ H^s=(1−\Delta)^{-s/2}L^2(\mathbb{R}^d), $$ $$ H^s_D=\{f\in H^s:f=0 \ a.e.\ on\ D^c\}. $$ Q: Is $C^\infty_c(D)$ dense in $H^s_D$(with norm $\|\cdot\|_{H^s}$) for any open set $D$?

Is there any element reference?

Density in fractional Sobolev space

Suppose $s∈(0,1)$, $D$ is an open set in $\mathbb{R}^d$. Define $$ H^s=(1−\Delta)^{-s/2}L^2\left(\mathbb{R}^d\right), $$ $$ H^s_D=\left\{f\in H^s:f=0 \mbox{ a.e. on } D^c\right\}. $$ Q: Is $C^\infty_c(D)$ dense in $H^s_D$(with norm $\|\cdot\|_{H^s}$) for any open set $D$?

Is there any element reference?

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paul garrett
paul garrett
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density in fractional DobolevSobolev space

density in fractional DobolevSobolev space

Suppose $s∈(0,1)$, $D$ is an open set in $\mathbb{R}^d$. Define $$ H^s=(1−\Delta)^{-s/2}L^2(\mathbb{R}^d), $$ $$ H^s_D=\{f\in H^s:f=0 \ a.e.\ on\ D^c\}. $$ Q: Is $C^\infty_c(D)$ dense in $H^s_D$(with norm $\|\cdot\|_{H^s}$) for any open set $D$?

Is there any element reference?

density in fractional Dobolev space

Suppose $s∈(0,1)$, $D$ is an open set in $\mathbb{R}^d$. Define $$ H^s=(1−\Delta)^{-s/2}L^2(\mathbb{R}^d), $$ $$ H^s_D=\{f\in H^s:f=0 \ a.e.\ on\ D^c\}. $$ Q: Is $C^\infty_c(D)$ dense in $H^s_D$(with norm $\|\cdot\|_{H^s}$) for any open set $D$?

Is there any element reference?

density in fractional Sobolev space

Suppose $s∈(0,1)$, $D$ is an open set in $\mathbb{R}^d$. Define $$ H^s=(1−\Delta)^{-s/2}L^2(\mathbb{R}^d), $$ $$ H^s_D=\{f\in H^s:f=0 \ a.e.\ on\ D^c\}. $$ Q: Is $C^\infty_c(D)$ dense in $H^s_D$(with norm $\|\cdot\|_{H^s}$) for any open set $D$?

Is there any element reference?

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Guohuan Zhao
Guohuan Zhao
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density in fractional Dobolev space

Suppose $s∈(0,1)$, $D$ is an open set in $\mathbb{R}^d$. Define $$ H^s=(1−\Delta)^{-s/2}L^2(\mathbb{R}^d), $$ $$ H^s_D=\{f\in H^s:f=0 \ a.e.\ on\ D^c\}. $$ Q: Is $C^\infty_c(D)$ dense in $H^s_D$(with norm $\|\cdot\|_{H^s}$) for any open set $D$?

Is there any element reference?