Concept guide
Boundary-value problem: checking the interior and all four edges separately for a Poisson–Dirichlet problem on a rectangle
On the rectangle Ω=(0,2)×(0,1) declare the BVP: in the interior Δu=u_xx+u_yy=4; Dirichlet data u(0,y)=y² on the left, u(2,y)=y²+2y+4 on the right (0≤y≤1), u(x,0)=x² on the bottom and u(x,1)=x²+x+1 on the top (0≤x≤2). For the quadratic polynomial candidates u=x²+xy+y² and v=2x²+xy−2x+y, record separately the interior Poisson obligation (a symbolic Laplacian identity) and the Dirichlet obligation on all four edges (exact restriction to each whole edge, then coefficient comparison); decide which candidate satisfies this declared BVP, and explain why v fails although its interior equation holds and it matches the data at all four corners.