I'd use the fact that a 2x2 orthogonal matrix acts as a reflection and/or a rotation and tackle this in the complex setting (the matrix acts as complex conjugation and/or multiplication by some unimodular complex number).
let f:R^2 -> R be harmonic and let A be a 2x2 orthogonal matrix. show that g(x) = f(Ax) is harmonic.
a function is harmonic if the laplacian of that function is 0. so i need to show that g_xx + g_yy = 0.
so [Dg(x)] = [Df'(Ax)][D(Ax)]=[f_x f_y][A] by the chain rule. and i let the matrix A have coefficients a11 = a, a12 = b, a21 = c, and a22 = d. therefore [Dg(x)] = [af_x + cf_y | bf_x + df_y] so the first partial of g with respect to x by the definition of the jacobian matrix is g_x = af_x + cf_y and g_xx would then be af_xx + cf_yy. similarly, i got g_yy = bf_xx + df_yy. so g_xx + g_yy = (a+b)f_xx + (c+d)f_yy. i know that f_xx + f_yy = 0 since it is given that f is harmonic. so i just need to show that (a+b) = (c+d). i tried using the information that A was orthogonal which means that the columns and rows are unit vectors and i ended up with a,b,c,d all equal to +/-1. which i'm not sure about since a+b can be 0 while c+d can be 2 and that would not work. have i done this problem correctly up to this point?
My mistake...I misread your question. What I said is the answer to a similar but different question.
Check out the Wikipedia article on orthogonal matrices--it shows that a 2x2 such matrix must have a special form that might make your calculations go easier.
based on the wikipedia article, there are 2 possibilities. the problem one possibility makes it so that g_xx + g_yy = 0 but the other possibility makes it so that g_xx + g_yy = f_xy + f_yx which may or may not be 0.
i made a mistake when i posted my attempt at the problem. g_xx = af_xx + cf_yx and g_yy = bf_yx + df_yy. is there still something i may be missing here?