Elliptic Regularity on Convex Domain












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$begingroup$


In many literature about elliptic regularity on convex domains, they impose that the domain should be polygonal. (In such cases, the inequality below holds)



However, it seems to me that the polygonal assumption is given to certify regularity of variational(weak) solution rather than inequality.



Question) For convex domain $Omega$ with piecewise smooth boundary, let $u$ be an $H^2$ solution of $Delta u=f$ with $u|_{partialOmega}=0$, does the following 'classical' inequality hold?



$$exists C,s.t.,|u|_{H^2}le C|f|_{H^0}$$






pde elliptic-equations elliptic-operators







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    In many literature about elliptic regularity on convex domains, they impose that the domain should be polygonal. (In such cases, the inequality below holds)



    However, it seems to me that the polygonal assumption is given to certify regularity of variational(weak) solution rather than inequality.



    Question) For convex domain $Omega$ with piecewise smooth boundary, let $u$ be an $H^2$ solution of $Delta u=f$ with $u|_{partialOmega}=0$, does the following 'classical' inequality hold?



    $$exists C,s.t.,|u|_{H^2}le C|f|_{H^0}$$






    pde elliptic-equations elliptic-operators







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      In many literature about elliptic regularity on convex domains, they impose that the domain should be polygonal. (In such cases, the inequality below holds)



      However, it seems to me that the polygonal assumption is given to certify regularity of variational(weak) solution rather than inequality.



      Question) For convex domain $Omega$ with piecewise smooth boundary, let $u$ be an $H^2$ solution of $Delta u=f$ with $u|_{partialOmega}=0$, does the following 'classical' inequality hold?



      $$exists C,s.t.,|u|_{H^2}le C|f|_{H^0}$$






      pde elliptic-equations elliptic-operators







      share|cite|improve this question









      $endgroup$




      In many literature about elliptic regularity on convex domains, they impose that the domain should be polygonal. (In such cases, the inequality below holds)



      However, it seems to me that the polygonal assumption is given to certify regularity of variational(weak) solution rather than inequality.



      Question) For convex domain $Omega$ with piecewise smooth boundary, let $u$ be an $H^2$ solution of $Delta u=f$ with $u|_{partialOmega}=0$, does the following 'classical' inequality hold?



      $$exists C,s.t.,|u|_{H^2}le C|f|_{H^0}$$






      pde elliptic-equations elliptic-operators




      pde elliptic-equations elliptic-operators






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 5 at 7:30









      C.ParkC.Park

      1568




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