For which $k in mathbbN$ and $p in[1,infty]$ is $u in W^k,p(B_1(0))$?

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Let $u: overline B_1(0)subsetmathbbR^2tomathbbR$ defined by

$$u(x_1,x_2)=x_1x_2(1-sqrtx_1^2+x_2^2)$$
For which $ k in mathbbN $ and $ p in[1,infty] $ is $ u in W^k,p(B_1(0)) $ ?
I don't have any clue. Thanks for helping !






pde sobolev-spaces







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  • 1




    Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
    – José Carlos Santos
    Sep 5 at 10:57










  • Your question will be easier to read (and write) if you only encase the math in dollar signs, not your entire question.
    – Arthur
    Sep 5 at 11:00







  • 2




    "I don't have any clue" is a pretty strong statement. Do you know what $W^k,p(B_1(0))$ is?
    – Umberto P.
    Sep 5 at 11:21










  • Yes , the Sobolev space . I think I have to compute the partial derivatives and look if they are in L^p
    – Matillo
    Sep 5 at 12:48










  • Yes. In fact $x_1x_2$ is (obviously?) in every Sobolev space $W^k,p(B_1(0))$ so you can just consider $x_1x_2 sqrtx_1^2 + x_2^2$. Start calculating.
    – Umberto P.
    Sep 5 at 14:36














up vote
0
down vote

favorite












Let $u: overline B_1(0)subsetmathbbR^2tomathbbR$ defined by

$$u(x_1,x_2)=x_1x_2(1-sqrtx_1^2+x_2^2)$$
For which $ k in mathbbN $ and $ p in[1,infty] $ is $ u in W^k,p(B_1(0)) $ ?
I don't have any clue. Thanks for helping !






pde sobolev-spaces







share|cite|improve this question



















  • 1




    Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
    – José Carlos Santos
    Sep 5 at 10:57










  • Your question will be easier to read (and write) if you only encase the math in dollar signs, not your entire question.
    – Arthur
    Sep 5 at 11:00







  • 2




    "I don't have any clue" is a pretty strong statement. Do you know what $W^k,p(B_1(0))$ is?
    – Umberto P.
    Sep 5 at 11:21










  • Yes , the Sobolev space . I think I have to compute the partial derivatives and look if they are in L^p
    – Matillo
    Sep 5 at 12:48










  • Yes. In fact $x_1x_2$ is (obviously?) in every Sobolev space $W^k,p(B_1(0))$ so you can just consider $x_1x_2 sqrtx_1^2 + x_2^2$. Start calculating.
    – Umberto P.
    Sep 5 at 14:36












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let $u: overline B_1(0)subsetmathbbR^2tomathbbR$ defined by

$$u(x_1,x_2)=x_1x_2(1-sqrtx_1^2+x_2^2)$$
For which $ k in mathbbN $ and $ p in[1,infty] $ is $ u in W^k,p(B_1(0)) $ ?
I don't have any clue. Thanks for helping !






pde sobolev-spaces







share|cite|improve this question















Let $u: overline B_1(0)subsetmathbbR^2tomathbbR$ defined by

$$u(x_1,x_2)=x_1x_2(1-sqrtx_1^2+x_2^2)$$
For which $ k in mathbbN $ and $ p in[1,infty] $ is $ u in W^k,p(B_1(0)) $ ?
I don't have any clue. Thanks for helping !






pde sobolev-spaces




pde sobolev-spaces






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share|cite|improve this question













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edited Sep 5 at 11:18









miracle173

7,16922247




7,16922247










asked Sep 5 at 10:51









Matillo

114




114







  • 1




    Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
    – José Carlos Santos
    Sep 5 at 10:57










  • Your question will be easier to read (and write) if you only encase the math in dollar signs, not your entire question.
    – Arthur
    Sep 5 at 11:00







  • 2




    "I don't have any clue" is a pretty strong statement. Do you know what $W^k,p(B_1(0))$ is?
    – Umberto P.
    Sep 5 at 11:21










  • Yes , the Sobolev space . I think I have to compute the partial derivatives and look if they are in L^p
    – Matillo
    Sep 5 at 12:48










  • Yes. In fact $x_1x_2$ is (obviously?) in every Sobolev space $W^k,p(B_1(0))$ so you can just consider $x_1x_2 sqrtx_1^2 + x_2^2$. Start calculating.
    – Umberto P.
    Sep 5 at 14:36












  • 1




    Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
    – José Carlos Santos
    Sep 5 at 10:57










  • Your question will be easier to read (and write) if you only encase the math in dollar signs, not your entire question.
    – Arthur
    Sep 5 at 11:00







  • 2




    "I don't have any clue" is a pretty strong statement. Do you know what $W^k,p(B_1(0))$ is?
    – Umberto P.
    Sep 5 at 11:21










  • Yes , the Sobolev space . I think I have to compute the partial derivatives and look if they are in L^p
    – Matillo
    Sep 5 at 12:48










  • Yes. In fact $x_1x_2$ is (obviously?) in every Sobolev space $W^k,p(B_1(0))$ so you can just consider $x_1x_2 sqrtx_1^2 + x_2^2$. Start calculating.
    – Umberto P.
    Sep 5 at 14:36







1




1




Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
– José Carlos Santos
Sep 5 at 10:57




Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
– José Carlos Santos
Sep 5 at 10:57












Your question will be easier to read (and write) if you only encase the math in dollar signs, not your entire question.
– Arthur
Sep 5 at 11:00





Your question will be easier to read (and write) if you only encase the math in dollar signs, not your entire question.
– Arthur
Sep 5 at 11:00





2




2




"I don't have any clue" is a pretty strong statement. Do you know what $W^k,p(B_1(0))$ is?
– Umberto P.
Sep 5 at 11:21




"I don't have any clue" is a pretty strong statement. Do you know what $W^k,p(B_1(0))$ is?
– Umberto P.
Sep 5 at 11:21












Yes , the Sobolev space . I think I have to compute the partial derivatives and look if they are in L^p
– Matillo
Sep 5 at 12:48




Yes , the Sobolev space . I think I have to compute the partial derivatives and look if they are in L^p
– Matillo
Sep 5 at 12:48












Yes. In fact $x_1x_2$ is (obviously?) in every Sobolev space $W^k,p(B_1(0))$ so you can just consider $x_1x_2 sqrtx_1^2 + x_2^2$. Start calculating.
– Umberto P.
Sep 5 at 14:36




Yes. In fact $x_1x_2$ is (obviously?) in every Sobolev space $W^k,p(B_1(0))$ so you can just consider $x_1x_2 sqrtx_1^2 + x_2^2$. Start calculating.
– Umberto P.
Sep 5 at 14:36















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