From his Word to Mathematica:
I suppose that y1 <= y2
A function U is written
u[s_] := Piecewise[
{{Integrate[f[s]^(1/n), {s,0,y}], 0<=s<=y1},
{Integrate[f[s]^(1/n), {s,0,y1}]+Integrate[-f[s]^(1/n), {s,y,y1}], y1<=s<=y2},
{Integrate[f[s]^(1/n), {s,0,y1}]+Integrate[-f[s]^(1/n), {s,y2,y1}]+Integrate[f[s]^(1/n), {s,y2,y}], y2<=s<=1}}]
such as
f[s_] := 1/2 (s^2 - (y1 + y2) s + y1*y2);
I think if I apply this condition
Integrate[u[s], {s, 0, 1}] == 0
I find
Integrate[Integrate[f[s]^(1/n), {s, 0, y}], {t, 0, y1}] +
Integrate[Integrate[f[s]^(1/n), {s, 0, y1}], {t, y1, y2}] +
Integrate[Integrate[(-f[s])^(1/n), {s, y, y1}], {t, y1, y2}] +
Integrate[Integrate[f[s]^(1/n), {s, 0, y1}], {t, y2, 1}] +
Integrate[Integrate[(-f[s])^(1/n), {s, y2, y1}], {t, y2, 1}] +
Integrate[Integrate[f[s]^(1/n), {s, y2, y}], {t, y2, 1}] == 0
I try to solve this equation to find a relationship between y1 and y2 with y1 and y2 are the roots of f[s]