Laplace equation on a disk

From the pictures you can see that the maximum and the minimum are attained at the boundary.

F[x_, y_] := Piecewise[{{2*y, 0 < y}, {0, y < 0}}]; 
sol = NDSolveValue[{Laplacian[u[x, y], {x, y}] == 0, DirichletCondition[
      u[x, y] == F[x, y], True]}, u, Element[{x, y}, Disk[]]]; 
Plot3D[sol[x, y], Element[{x, y}, Disk[]]]

G[x_, y_] := Piecewise[{{2*x^2,  x>0}, {0, x < 0}}]; 
sol2 = NDSolveValue[{Laplacian[u[x, y], {x, y}] == 0, DirichletCondition[
      u[x, y] == G[x, y], True]}, u, Element[{x, y}, Disk[]]]; 
Plot3D[sol2[x, y], Element[{x, y}, Disk[]]]

NDSolve[{Laplacian[u[x,y],{x,y}]==Sin[5*x]*Cos[5*y],DirichletCondition[u[x,y]==Cos[10*ArcTan[(y/x)]],True]},u,{x,y}∈Disk[]]

Plot3D[Evaluate[u[x,y]/.%],{x,y}∈Disk[],PlotRange->All ]

NDSolveValue[{(Subscript[∇, {x,y}]^2)u[x,y]==1,DirichletCondition[u[x,y]==0,True]},u,{x,y}∈Disk[]]

Plot3D[%[x,y],{x,y}∈Disk[]]

NDSolveValue[{(Subscript[∇, {x,y}]^2)u[x,y]==x^2+y^2,DirichletCondition[u[x,y]==0,True]},u,{x,y}∈Disk[]]

Plot3D[%[x,y],{x,y}∈Disk[]]

NDSolveValue[{(Subscript[∇, {x,y}]^2)u[x,y]==0,DirichletCondition[u[x,y]==Sin[x y],True]},u,{x,y}∈Disk[]]

Plot3D[%[x,y],{x,y}∈Disk[]]

NDSolveValue[{(Subscript[∇, {x,y}]^2)u[x,y]==-x^2-y^2,DirichletCondition[u[x,y]==0,True]},u,{x,y}∈Disk[]]

Plot3D[%[x,y],{x,y}∈Disk[]]