Skip to main content
MathOverflow

Return to Question

added 6 characters in body
Source Link
Spal
Spal
  • 99
  • 1
  • 7

Let $s\in (0,1)$ and $u\in D^{s, 2} (\mathbb R^N)$, $\phi\in H^s_{0}(B).$ Here $B$ is a unit ball in $\mathbb R^N$ and $$H^s_{0}(B)=\{ u\in H^{s} (\mathbb R^N): u=0 \text{ in } \mathbb R^N- B\}$$$$H^s_{0}(B)=\{ \phi\in H^{s} (\mathbb R^N): \phi=0 \text{ in } \mathbb R^N- B\}$$ and $$D^{s, 2} (\mathbb R^N)= \bigg\{u\in L^{\frac{2N}{N-2s}}(\mathbb R^N): \int_{\mathbb R^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dx dy<+\infty\bigg\}.$$

Is integration by parts valid for

$$\int_{B} (-\Delta)^s u \phi dx= \int_{\mathbb R^{N}} (-\Delta)^{s/2} u (-\Delta)^{s/2} \phi dx.$$

References are welcome.

Let $s\in (0,1)$ and $u\in D^{s, 2} (\mathbb R^N)$, $\phi\in H^s_{0}(B).$ Here $B$ is a unit ball in $\mathbb R^N$ and $$H^s_{0}(B)=\{ u\in H^{s} (\mathbb R^N): u=0 \text{ in } \mathbb R^N- B\}$$ and $$D^{s, 2} (\mathbb R^N)= \bigg\{u\in L^{\frac{2N}{N-2s}}(\mathbb R^N): \int_{\mathbb R^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dx dy<+\infty\bigg\}.$$

Is integration by parts valid for

$$\int_{B} (-\Delta)^s u \phi dx= \int_{\mathbb R^{N}} (-\Delta)^{s/2} u (-\Delta)^{s/2} \phi dx.$$

References are welcome.

Let $s\in (0,1)$ and $u\in D^{s, 2} (\mathbb R^N)$, $\phi\in H^s_{0}(B).$ Here $B$ is a unit ball in $\mathbb R^N$ and $$H^s_{0}(B)=\{ \phi\in H^{s} (\mathbb R^N): \phi=0 \text{ in } \mathbb R^N- B\}$$ and $$D^{s, 2} (\mathbb R^N)= \bigg\{u\in L^{\frac{2N}{N-2s}}(\mathbb R^N): \int_{\mathbb R^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dx dy<+\infty\bigg\}.$$

Is integration by parts valid for

$$\int_{B} (-\Delta)^s u \phi dx= \int_{\mathbb R^{N}} (-\Delta)^{s/2} u (-\Delta)^{s/2} \phi dx.$$

References are welcome.

added 18 characters in body
Source Link
Spal
Spal
  • 99
  • 1
  • 7

Let $s\in (0,1)$ and $u\in D^{s, 2} (\mathbb R^N)$, $\phi\in H^s_{0}(B).$ Here $B$ is a unit ball in $\mathbb R^N$ and $$H^s_{0}(B)=\{ u\in H^{s} (\mathbb R^N): u=0 \text{ in } \mathbb R^N- B\}$$ and $$D^{s, 2} (\mathbb R^N)= \bigg\{u\in L^{\frac{2N}{N-2s}}(\mathbb R^N): \int_{\mathbb R^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dx dy<+\infty\bigg\}.$$

Is integration by parts valid for

$$\int_{B} (-\Delta)^s u \phi= \int_{B} (-\Delta)^{s/2} u (-\Delta)^{s/2} \phi.$$$$\int_{B} (-\Delta)^s u \phi dx= \int_{\mathbb R^{N}} (-\Delta)^{s/2} u (-\Delta)^{s/2} \phi dx.$$

References are welcome.

Let $s\in (0,1)$ and $u\in D^{s, 2} (\mathbb R^N)$, $\phi\in H^s_{0}(B).$ Here $B$ is a unit ball in $\mathbb R^N$ and $$H^s_{0}(B)=\{ u\in H^{s} (\mathbb R^N): u=0 \text{ in } \mathbb R^N- B\}$$ and $$D^{s, 2} (\mathbb R^N)= \bigg\{u\in L^{\frac{2N}{N-2s}}(\mathbb R^N): \int_{\mathbb R^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dx dy<+\infty\bigg\}.$$

Is integration by parts valid for

$$\int_{B} (-\Delta)^s u \phi= \int_{B} (-\Delta)^{s/2} u (-\Delta)^{s/2} \phi.$$

References are welcome.

Let $s\in (0,1)$ and $u\in D^{s, 2} (\mathbb R^N)$, $\phi\in H^s_{0}(B).$ Here $B$ is a unit ball in $\mathbb R^N$ and $$H^s_{0}(B)=\{ u\in H^{s} (\mathbb R^N): u=0 \text{ in } \mathbb R^N- B\}$$ and $$D^{s, 2} (\mathbb R^N)= \bigg\{u\in L^{\frac{2N}{N-2s}}(\mathbb R^N): \int_{\mathbb R^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dx dy<+\infty\bigg\}.$$

Is integration by parts valid for

$$\int_{B} (-\Delta)^s u \phi dx= \int_{\mathbb R^{N}} (-\Delta)^{s/2} u (-\Delta)^{s/2} \phi dx.$$

References are welcome.

added 158 characters in body
Source Link
Spal
Spal
  • 99
  • 1
  • 7

Let $s\in (0,1)$ and $u\in D^{s, 2} (\mathbb R^N)$, $\phi\in H^s_{0}(B).$ Here $B$ is a unit ball in $\mathbb R^N$ and $$H^s_{0}(B)=\{ u\in H^{s} (\mathbb R^N): u=0 \text{ in } \mathbb R^N- B\}.$$$$H^s_{0}(B)=\{ u\in H^{s} (\mathbb R^N): u=0 \text{ in } \mathbb R^N- B\}$$ and $$D^{s, 2} (\mathbb R^N)= \bigg\{u\in L^{\frac{2N}{N-2s}}(\mathbb R^N): \int_{\mathbb R^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dx dy<+\infty\bigg\}.$$

Is integration by parts valid for

$$\int_{B} (-\Delta)^s u \phi= \int_{B} (-\Delta)^{s/2} u (-\Delta)^{s/2} \phi.$$

References are welcome.

Let $s\in (0,1)$ and $u\in D^{s, 2} (\mathbb R^N)$, $\phi\in H^s_{0}(B).$ Here $B$ is a unit ball in $\mathbb R^N$ and $$H^s_{0}(B)=\{ u\in H^{s} (\mathbb R^N): u=0 \text{ in } \mathbb R^N- B\}.$$

Is integration by parts valid for

$$\int_{B} (-\Delta)^s u \phi= \int_{B} (-\Delta)^{s/2} u (-\Delta)^{s/2} \phi.$$

References are welcome.

Let $s\in (0,1)$ and $u\in D^{s, 2} (\mathbb R^N)$, $\phi\in H^s_{0}(B).$ Here $B$ is a unit ball in $\mathbb R^N$ and $$H^s_{0}(B)=\{ u\in H^{s} (\mathbb R^N): u=0 \text{ in } \mathbb R^N- B\}$$ and $$D^{s, 2} (\mathbb R^N)= \bigg\{u\in L^{\frac{2N}{N-2s}}(\mathbb R^N): \int_{\mathbb R^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dx dy<+\infty\bigg\}.$$

Is integration by parts valid for

$$\int_{B} (-\Delta)^s u \phi= \int_{B} (-\Delta)^{s/2} u (-\Delta)^{s/2} \phi.$$

References are welcome.

Source Link
Spal
Spal
  • 99
  • 1
  • 7