Domain of Fractional power of Operator

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I have some questions about the Domain of the fractional power of operator as follows. Let $Omega$ be a bounded domain of $mathbbR^n$ with regular boundary. Consider the operator $C=Delta_N- e^2$, where $Delta_N$ is the Neumann realization of Laplace operator in $H=L^2(Omega,mathbbR)$ and $ein H$ is Lipschitz function. Let $E=C(barOmega, mathbbR)$ and $P_alpha$ be the Domain of $(-C_E)^alpha$, for some $alpha in (0,1).$ Then some following properties is true or not, if it is true how can we prove that?



  1. Let $varphi colon mathbbR to mathbbR$ is continuously differentiable and $f:E to E$, $f(u)=varphi(u(x))$ then $f(P_alpha)$ is contained by $P_alpha$ ?

  2. If $u,v in P_alpha$, what about $uv$?

Thank you so much for your attention.




functional-analysis pde




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    up vote
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    I have some questions about the Domain of the fractional power of operator as follows. Let $Omega$ be a bounded domain of $mathbbR^n$ with regular boundary. Consider the operator $C=Delta_N- e^2$, where $Delta_N$ is the Neumann realization of Laplace operator in $H=L^2(Omega,mathbbR)$ and $ein H$ is Lipschitz function. Let $E=C(barOmega, mathbbR)$ and $P_alpha$ be the Domain of $(-C_E)^alpha$, for some $alpha in (0,1).$ Then some following properties is true or not, if it is true how can we prove that?



    1. Let $varphi colon mathbbR to mathbbR$ is continuously differentiable and $f:E to E$, $f(u)=varphi(u(x))$ then $f(P_alpha)$ is contained by $P_alpha$ ?

    2. If $u,v in P_alpha$, what about $uv$?

    Thank you so much for your attention.




    functional-analysis pde




    share|cite|improve this question























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I have some questions about the Domain of the fractional power of operator as follows. Let $Omega$ be a bounded domain of $mathbbR^n$ with regular boundary. Consider the operator $C=Delta_N- e^2$, where $Delta_N$ is the Neumann realization of Laplace operator in $H=L^2(Omega,mathbbR)$ and $ein H$ is Lipschitz function. Let $E=C(barOmega, mathbbR)$ and $P_alpha$ be the Domain of $(-C_E)^alpha$, for some $alpha in (0,1).$ Then some following properties is true or not, if it is true how can we prove that?



      1. Let $varphi colon mathbbR to mathbbR$ is continuously differentiable and $f:E to E$, $f(u)=varphi(u(x))$ then $f(P_alpha)$ is contained by $P_alpha$ ?

      2. If $u,v in P_alpha$, what about $uv$?

      Thank you so much for your attention.




      functional-analysis pde




      share|cite|improve this question













      I have some questions about the Domain of the fractional power of operator as follows. Let $Omega$ be a bounded domain of $mathbbR^n$ with regular boundary. Consider the operator $C=Delta_N- e^2$, where $Delta_N$ is the Neumann realization of Laplace operator in $H=L^2(Omega,mathbbR)$ and $ein H$ is Lipschitz function. Let $E=C(barOmega, mathbbR)$ and $P_alpha$ be the Domain of $(-C_E)^alpha$, for some $alpha in (0,1).$ Then some following properties is true or not, if it is true how can we prove that?



      1. Let $varphi colon mathbbR to mathbbR$ is continuously differentiable and $f:E to E$, $f(u)=varphi(u(x))$ then $f(P_alpha)$ is contained by $P_alpha$ ?

      2. If $u,v in P_alpha$, what about $uv$?

      Thank you so much for your attention.





      functional-analysis pde





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Jul 22 at 21:07









      user539887

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      asked Jul 22 at 20:12









      Nhu Nguyen

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