Problem 6. Three harmonic functions uk,k=1,2,3 are defined on the domain kR2, k=1,2,3 given as follows. Particularly, 1 is a disk of radius 1,2 is a square of side length 2 , and 3 is an equilateral triangle of side length 2 . (d) 341 (b) S2 On the boundary each function uk takes two values, 0 or 1 . As indicated in the graph above, uk(x,y)=0 for all (x,y) on the dashed portion, whereas uk(x,y)=1 for all (x,y) on the solid portion. (a) Which k{1,2,3} has the largest value uk(xk,yk), where (xk,yk) is the (rotational) center of the domain k ? (b) Which k{1,2,3} has the largest integral kuk(x,y)dxdy ? Problem 6. Three harmonic functions uk,k=1,2,3 are defined on the domain kR2, k=1,2,3 given as follows. Particularly, 1 is a disk of radius 1,2 is a square of side length 2 , and 3 is an equilateral triangle of side length 2 . (d) 341 (b) S2 On the boundary each function uk takes two values, 0 or 1 . As indicated in the graph above, uk(x,y)=0 for all (x,y) on the dashed portion, whereas uk(x,y)=1 for all (x,y) on the solid portion. (a) Which k{1,2,3} has the largest value uk(xk,yk), where (xk,yk) is the (rotational) center of the domain k ? (b) Which k{1,2,3} has the largest integral kuk(x,y)dxdy