Kernel of a bilinear form - Structural Mechanics

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If I have a bilinear form



$$a : (H^1(Omega))^3 times (H^1(Omega))^3 mapsto mathbbR hspace0.9in a(vecu, vecv) = int_Omega(Dvecu)^T C (Dvecv), dOmega $$



I would like to find the kernel of this bilinear form when $u in V$ such that,
$$V = u_x(0,0,0) = u_y(0,0,0) = u_z(0,0,0) = 0 $$



I would like to comment what are C and D operators.
D is Symmetric derivative operator.
$$Dvecu = beginbmatrix fracpartial u_xpartial x \ fracpartial u_ypartial y \ fracpartial u_zpartial z \ frac12( fracpartial u_xpartial y + fracpartial u_ypartial x) \ frac12( fracpartial u_ypartial z + fracpartial u_zpartial y) \ frac12( fracpartial u_xpartial z + fracpartial u_zpartial x)endbmatrix : Omega mapsto mathbbR^6$$
C is fourth order isotropic elasticity tensor in voigt notation.
http://web.mit.edu/16.20/homepage/3_Constitutive/Constitutive_files/module_3_with_solutions.pdf
pg no - 18




functional-analysis bilinear-form




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    up vote
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    down vote

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    If I have a bilinear form



    $$a : (H^1(Omega))^3 times (H^1(Omega))^3 mapsto mathbbR hspace0.9in a(vecu, vecv) = int_Omega(Dvecu)^T C (Dvecv), dOmega $$



    I would like to find the kernel of this bilinear form when $u in V$ such that,
    $$V = u_x(0,0,0) = u_y(0,0,0) = u_z(0,0,0) = 0 $$



    I would like to comment what are C and D operators.
    D is Symmetric derivative operator.
    $$Dvecu = beginbmatrix fracpartial u_xpartial x \ fracpartial u_ypartial y \ fracpartial u_zpartial z \ frac12( fracpartial u_xpartial y + fracpartial u_ypartial x) \ frac12( fracpartial u_ypartial z + fracpartial u_zpartial y) \ frac12( fracpartial u_xpartial z + fracpartial u_zpartial x)endbmatrix : Omega mapsto mathbbR^6$$
    C is fourth order isotropic elasticity tensor in voigt notation.
    http://web.mit.edu/16.20/homepage/3_Constitutive/Constitutive_files/module_3_with_solutions.pdf
    pg no - 18




    functional-analysis bilinear-form




    share|cite|improve this question





















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      If I have a bilinear form



      $$a : (H^1(Omega))^3 times (H^1(Omega))^3 mapsto mathbbR hspace0.9in a(vecu, vecv) = int_Omega(Dvecu)^T C (Dvecv), dOmega $$



      I would like to find the kernel of this bilinear form when $u in V$ such that,
      $$V = u_x(0,0,0) = u_y(0,0,0) = u_z(0,0,0) = 0 $$



      I would like to comment what are C and D operators.
      D is Symmetric derivative operator.
      $$Dvecu = beginbmatrix fracpartial u_xpartial x \ fracpartial u_ypartial y \ fracpartial u_zpartial z \ frac12( fracpartial u_xpartial y + fracpartial u_ypartial x) \ frac12( fracpartial u_ypartial z + fracpartial u_zpartial y) \ frac12( fracpartial u_xpartial z + fracpartial u_zpartial x)endbmatrix : Omega mapsto mathbbR^6$$
      C is fourth order isotropic elasticity tensor in voigt notation.
      http://web.mit.edu/16.20/homepage/3_Constitutive/Constitutive_files/module_3_with_solutions.pdf
      pg no - 18




      functional-analysis bilinear-form




      share|cite|improve this question











      If I have a bilinear form



      $$a : (H^1(Omega))^3 times (H^1(Omega))^3 mapsto mathbbR hspace0.9in a(vecu, vecv) = int_Omega(Dvecu)^T C (Dvecv), dOmega $$



      I would like to find the kernel of this bilinear form when $u in V$ such that,
      $$V = u_x(0,0,0) = u_y(0,0,0) = u_z(0,0,0) = 0 $$



      I would like to comment what are C and D operators.
      D is Symmetric derivative operator.
      $$Dvecu = beginbmatrix fracpartial u_xpartial x \ fracpartial u_ypartial y \ fracpartial u_zpartial z \ frac12( fracpartial u_xpartial y + fracpartial u_ypartial x) \ frac12( fracpartial u_ypartial z + fracpartial u_zpartial y) \ frac12( fracpartial u_xpartial z + fracpartial u_zpartial x)endbmatrix : Omega mapsto mathbbR^6$$
      C is fourth order isotropic elasticity tensor in voigt notation.
      http://web.mit.edu/16.20/homepage/3_Constitutive/Constitutive_files/module_3_with_solutions.pdf
      pg no - 18





      functional-analysis bilinear-form





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