Optimization problem over integration sublevel set in $mathbbR^n$

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Suppose $phi:mathbb R^ntomathbb R^n$ is a fixed smooth vector field; if it's useful, we can assume $|phi(x)|_2leq1$ for all $xinmathbb R^n$. Consider the following optimization problem parameterized by $t$:
$$
f_phi(t):=
sup_lambdainmathbb Rleft[
lambda t - int_Omegamax(0,lambda-nablacdotphi(x)),dx
right].
$$
Can we write an explicit expression for $f_phi(t)$ in terms of the function $phi$ and scalar $t$?



I assume the best way to approach this problem is using the Reynolds Transport Theorem, but the computation seemed to get messy quickly! In a sense, the integral term can also be understood as the integral of $lambda-nablacdotphi$ over the set $x:lambdageqnablacdotphi(x)$.







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    up vote
    0
    down vote

    favorite












    Suppose $phi:mathbb R^ntomathbb R^n$ is a fixed smooth vector field; if it's useful, we can assume $|phi(x)|_2leq1$ for all $xinmathbb R^n$. Consider the following optimization problem parameterized by $t$:
    $$
    f_phi(t):=
    sup_lambdainmathbb Rleft[
    lambda t - int_Omegamax(0,lambda-nablacdotphi(x)),dx
    right].
    $$
    Can we write an explicit expression for $f_phi(t)$ in terms of the function $phi$ and scalar $t$?



    I assume the best way to approach this problem is using the Reynolds Transport Theorem, but the computation seemed to get messy quickly! In a sense, the integral term can also be understood as the integral of $lambda-nablacdotphi$ over the set $x:lambdageqnablacdotphi(x)$.







    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Suppose $phi:mathbb R^ntomathbb R^n$ is a fixed smooth vector field; if it's useful, we can assume $|phi(x)|_2leq1$ for all $xinmathbb R^n$. Consider the following optimization problem parameterized by $t$:
      $$
      f_phi(t):=
      sup_lambdainmathbb Rleft[
      lambda t - int_Omegamax(0,lambda-nablacdotphi(x)),dx
      right].
      $$
      Can we write an explicit expression for $f_phi(t)$ in terms of the function $phi$ and scalar $t$?



      I assume the best way to approach this problem is using the Reynolds Transport Theorem, but the computation seemed to get messy quickly! In a sense, the integral term can also be understood as the integral of $lambda-nablacdotphi$ over the set $x:lambdageqnablacdotphi(x)$.







      share|cite|improve this question














      Suppose $phi:mathbb R^ntomathbb R^n$ is a fixed smooth vector field; if it's useful, we can assume $|phi(x)|_2leq1$ for all $xinmathbb R^n$. Consider the following optimization problem parameterized by $t$:
      $$
      f_phi(t):=
      sup_lambdainmathbb Rleft[
      lambda t - int_Omegamax(0,lambda-nablacdotphi(x)),dx
      right].
      $$
      Can we write an explicit expression for $f_phi(t)$ in terms of the function $phi$ and scalar $t$?



      I assume the best way to approach this problem is using the Reynolds Transport Theorem, but the computation seemed to get messy quickly! In a sense, the integral term can also be understood as the integral of $lambda-nablacdotphi$ over the set $x:lambdageqnablacdotphi(x)$.









      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Aug 12 at 2:16









      David G. Stork

      8,05421131




      8,05421131










      asked Aug 12 at 2:00









      Justin Solomon

      176113




      176113

























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