I would like to find a continuous function $u:[0,1]\to \mathbb{R}$, which is a minimizer of the following functional $$ F(u) = \frac{1}{2}\int_0^1 \int_0^1 \left( \frac{u(x)-u(y)}{x-y}\right)^2\mathrm{d}x \, \mathrm{d}y $$ with boundary values $u(0) = 0$ and $u(1) = 1$. This functional is physically motivated (in peridynamics). It is easy to find the equation of optimality: $$ 0 = \int_0^1 \frac{u(y) - u(x)}{(y-x)^2} \mathrm{d}y$$ for almost every $x$ (here the integral should be understood in the sense of principal value). But I cannot solve this integral equation for $u$. Is it possible to solve such a problem, preferably by some explicit formula or by using some numerical method?
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