Mapping the FindMinimum function over a matrix

Question

I'm quite new to Mathematica and so still trying to work out how to use some of its syntax and functions. I have a function for the free energy of a generic system with two order parameters. I am then creating a table with the constants r and g ranging from -100 to 100 in steps of 1. This is shown below in the code:

clear[func];
func[r_, g_, u1_, u2_, u12_, \[Phi]1_, \[Phi]2_] := 1/2 r (\[Phi]1^2 + \[Phi]2^2) + 1/2 g (\[Phi]2^2 - \[Phi]1^2) + u1 \[Phi]1^4 + u2 \[Phi]2^4 + u12 \[Phi]1^2 \[Phi]2^2
regime1 = Table[{func[r, g, 10, 10, 50, \[Phi]1, \[Phi]2], r, g, \[Phi]1, \[Phi]2}, {r, -100, 100, 1}, {g, -100, 100, 1}]

I then get a 2D matrix with each element as a list. This list consists of: the function with the set of parameters passed therein, and then elements for the value of r and the value of g that was passed.

I wanted to map over the first element in each list in the matrix so as to find the minimum values of /[Phi]1 and /[Phi]2 using the /@ syntax and wanted to use the FindMinimum function. I was wondering whether anyone could help me to do this.

Answer 1

n = 10;
regime1 = Table[{func[r, g, 10, 10, 50, ϕ1, ϕ2], r,g, ϕ1, ϕ2}, {r, -n, n, 1}, {g, -n, n, 1}];

f = X \[Function] {X[[2]], X[[3]]} -> FindMinimum[X[[1]], Evaluate[X[[4]]], Evaluate[X[[5]]]];
results = Map[f, regime1, {2}];