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Hi,

Be careful, I guess what is done in the Evan's PDE book is for $W^{1,2}_0(\Omega)$ functions. If you look at all $W^{1,2}(\Omega)$, $\mathscr{D}(\Omega)$ (test functions) is not a dense subspace. Hence you may not inject the dual of $W^{1,2}(\Omega)$ in the space of distributions.

Your formula $f=f_0 -\sum_{i=1}^N \partial_{x_i} f_i$ may hence be misleading, no ?

Nevertheless, identifying $W^{1,p}(\Omega)$ with a " pair " of $L^p(\Omega)$ spaces, you may prove that its dual consists of a " pair " of $L^{p'}(\Omega)$ spaces (for finite values of $p$ of course). This approach is detailed in the book of Adams [*], p.62.

For the fractionnal case, I don't know about their duals. But I guess that your expected formula (which should be $f=f_0+(-\Delta)^{s/2} f_1$, no ?) would make sense only in the distribution framework (if not, I don't understand what is your definition of the fractionnal laplacian) : such a description (if it does exist) should hold only for elements of $W^{s,p}_0(\Omega)'$ .

[*] See here : http://www.amazon.com/Sobolev-Spaces-Second-Applied-Mathematics/dp/0120441438

Hi,

I guess what is done in the Evan's PDE book is for $W^{1,2}_0(\Omega)$ functions, right ? If you look at all $W^{1,2}(\Omega)$, $\mathscr{D}(\Omega)$ (test functions) is not a dense subspace. Hence you may not inject the dual of $W^{1,2}(\Omega)$ in the space of distributions.

Your formula $f=f_0 -\sum_{i=1}^N \partial_{x_i} f_i$ may then be misleading, no ?

Nevertheless, identifying $W^{1,p}(\Omega)$ with a " pair " of $L^p(\Omega)$ spaces, you may prove that its dual consists of a " pair " of $L^{p'}(\Omega)$ spaces (for finite values of $p$ of course). This approach is detailed in the book of Adams [*], p.62.

For the fractionnal case, I don't know about their duals. But I guess that your expected formula (which should be $f=f_0+(-\Delta)^{s/2} f_1$, no ?) would make sense only in the distribution framework (if not, I don't understand what is your definition of the fractionnal laplacian) : such a description (if it does exist) should hold only for elements of $W^{s,p}_0(\Omega)'$ .

[*] See here : http://www.amazon.com/Sobolev-Spaces-Second-Applied-Mathematics/dp/0120441438

Hi.

Be careful, I guess what is done in the Evan's PDE book is for $W^{1,2}_0(\Omega)$ functions. If you look at all $W^{1,2}(\Omega)$, $\mathscr{D}(\Omega)$ (test functions) is not a dense subspace. Hence you may not inject the dual of $W^{1,2}(\Omega)$ in the space of distributions.

Your formula $f=f_0 -\sum_{i=1}^N \partial_{x_i} f_i$ may hence be misleading.

Nevertheless, identifying $W^{1,p}(\Omega)$ with a " pair " of $L^p(\Omega)$ spaces, you may prove that its dual consists of a " pair " of $L^{p'}(\Omega)$ spaces (for finite values of $p$ of course). This approach is detailed in the book of Adams [*], p.62.

For the fractionnal case, I don't know about their duals. But I guess that your expected formula (which should be $f=f_0+(-\Delta)^{s/2} f_1$, no ?) would make sense only in the distribution framework (if not, I don't understand what is your definition of the fractionnal laplacian) : such a description (if it does exist) should hold only for elements of $W^{s,p}_0(\Omega)'$ .

[*] See here : http://www.amazon.com/Sobolev-Spaces-Second-Applied-Mathematics/dp/0120441438

Hi,

Be careful, I guess what is done in the Evan's PDE book is for $W^{1,2}_0(\Omega)$ functions. If you look at all $W^{1,2}(\Omega)$, $\mathscr{D}(\Omega)$ (test functions) is not a dense subspace. Hence you may not inject the dual of $W^{1,2}(\Omega)$ in the space of distributions.

Your formula $f=f_0 -\sum_{i=1}^N \partial_{x_i} f_i$ may hence be misleading, no ?

Nevertheless, identifying $W^{1,p}(\Omega)$ with a " pair " of $L^p(\Omega)$ spaces, you may prove that its dual consists of a " pair " of $L^{p'}(\Omega)$ spaces (for finite values of $p$ of course). This approach is detailed in the book of Adams [*], p.62.

For the fractionnal case, I don't know about their duals. But I guess that your expected formula (which should be $f=f_0+(-\Delta)^{s/2} f_1$, no ?) would make sense only in the distribution framework (if not, I don't understand what is your definition of the fractionnal laplacian) : such a description (if it does exist) should hold only for elements of $W^{s,p}_0(\Omega)'$ .

[*] See here : http://www.amazon.com/Sobolev-Spaces-Second-Applied-Mathematics/dp/0120441438

Hi.

Be careful, I guess what is done in the Evan's PDE book is for $W^{1,2}_0(\Omega)$ functions. If you look at all $W^{1,2}(\Omega)$, $\mathscr{D}(\Omega)$ (test functions) is not a dense subspace. Hence you may not inject the dual of $W^{1,2}(\Omega)$ in the space of distributions.

Your formula $f=f_0 -\sum_{i=1}^N \partial_{x_i} f_i$ may hence be misleading.

Nevertheless, identifying $W^{1,p}(\Omega)$ with a " pair " of $L^p(\Omega)$ spaces, you may prove that its dual consists of a " pair " of $L^{p'}(\Omega)$ spaces (for finite values of $p$ of course). This approach is detailed in the book of Adams [*], p.62.

For the fractionnal case, I don't know about their duals.

[*] See here : http://www.amazon.com/Sobolev-Spaces-Second-Applied-Mathematics/dp/0120441438

Hi.

Be careful, I guess what is done in the Evan's PDE book is for $W^{1,2}_0(\Omega)$ functions. If you look at all $W^{1,2}(\Omega)$, $\mathscr{D}(\Omega)$ (test functions) is not a dense subspace. Hence you may not inject the dual of $W^{1,2}(\Omega)$ in the space of distributions.

Your formula $f=f_0 -\sum_{i=1}^N \partial_{x_i} f_i$ may hence be misleading.

Nevertheless, identifying $W^{1,p}(\Omega)$ with a " pair " of $L^p(\Omega)$ spaces, you may prove that its dual consists of a " pair " of $L^{p'}(\Omega)$ spaces (for finite values of $p$ of course). This approach is detailed in the book of Adams [*], p.62.

For the fractionnal case, I don't know about their duals. But I guess that your expected formula (which should be $f=f_0+(-\Delta)^{s/2} f_1$, no ?) would make sense only in the distribution framework (if not, I don't understand what is your definition of the fractionnal laplacian) : such a description (if it does exist) should hold only for elements of $W^{s,p}_0(\Omega)'$ .

[*] See here : http://www.amazon.com/Sobolev-Spaces-Second-Applied-Mathematics/dp/0120441438