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Find the Euler equation for the functional

J(u)=\iint \limits _{G}\left[u_{x}^{2}+u_{y}^{2}+2f(x,y)u(x,y)\right]dxdy\,

where G\, is a closed region in the xy\, plane and u\, has continuous second partial derivatives.

L_{u}-{\frac  {\partial }{\partial x}}L_{{u_{x}}}-{\frac  {\partial }{\partial y}}L_{{u_{y}}} =0\,
2f(x,y)-{\frac  {\partial }{\partial x}}(2u_{x})-{\frac  {\partial }{\partial y}}(2u_{y}) =0\,
2f(x,y)-2u_{{xx}}-2u_{{yy}}\, =0\,
f(x,y)\, =u_{{xx}}+u_{{yy}}\,


Main Page : Calculus of Variations