mollification with boundary condition

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Consider a function $f:mathbbD^n to mathbbR$, where $mathbbD^n = x$, with smoothness in $int(mathbbD^n)$ and on $partial(mathbbD^n)=mathbbS^n$. That is, $f|_int(mathbbD^n) $ and $f|_mathbbS^n$ are smooth, but may failed to have a $fracpartial fpartial n$ for the outer normal vector on the boundary.



Now my question is, can we mollify such a function $f$ to be a smooth or just $C^1(mathbbD^n)$, by a sequence of function $f_k,$, with the boundary condition $f_k|_mathbbS^n=f|_mathbbS^n$ ?



Great thanks!




real-analysis




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    Consider a function $f:mathbbD^n to mathbbR$, where $mathbbD^n = x$, with smoothness in $int(mathbbD^n)$ and on $partial(mathbbD^n)=mathbbS^n$. That is, $f|_int(mathbbD^n) $ and $f|_mathbbS^n$ are smooth, but may failed to have a $fracpartial fpartial n$ for the outer normal vector on the boundary.



    Now my question is, can we mollify such a function $f$ to be a smooth or just $C^1(mathbbD^n)$, by a sequence of function $f_k,$, with the boundary condition $f_k|_mathbbS^n=f|_mathbbS^n$ ?



    Great thanks!




    real-analysis




    share|cite|improve this question





















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      down vote

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      Consider a function $f:mathbbD^n to mathbbR$, where $mathbbD^n = x$, with smoothness in $int(mathbbD^n)$ and on $partial(mathbbD^n)=mathbbS^n$. That is, $f|_int(mathbbD^n) $ and $f|_mathbbS^n$ are smooth, but may failed to have a $fracpartial fpartial n$ for the outer normal vector on the boundary.



      Now my question is, can we mollify such a function $f$ to be a smooth or just $C^1(mathbbD^n)$, by a sequence of function $f_k,$, with the boundary condition $f_k|_mathbbS^n=f|_mathbbS^n$ ?



      Great thanks!




      real-analysis




      share|cite|improve this question











      Consider a function $f:mathbbD^n to mathbbR$, where $mathbbD^n = x$, with smoothness in $int(mathbbD^n)$ and on $partial(mathbbD^n)=mathbbS^n$. That is, $f|_int(mathbbD^n) $ and $f|_mathbbS^n$ are smooth, but may failed to have a $fracpartial fpartial n$ for the outer normal vector on the boundary.



      Now my question is, can we mollify such a function $f$ to be a smooth or just $C^1(mathbbD^n)$, by a sequence of function $f_k,$, with the boundary condition $f_k|_mathbbS^n=f|_mathbbS^n$ ?



      Great thanks!





      real-analysis





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      asked Jul 30 at 1:24









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