I have the following calculus of variations problem:
$$\min_{f(\cdot)}\int_0^1(f(x)+g(x))^\phi dx \quad \text { s.t. } \int_0^1f(x)dx=A, \quad f(x)\geq 0 \quad \forall x\in [0,1]$$
For some given function $g(x)\geq0$ and a given constant $A>0$. $g(x)$ satisfies the following conditions:
- $g(x)=0$ for $x\in [0,a]$ for some $a\in[0,1]$.
- $g'(x) >0$ on $(a,1)$
- Hence $g(x)\geq 0$
I try to apply the Kuhn Tucker theorem and formulate the lagrangian:
$$\int_0^1\left[(f(x)+g(x))^\phi -\lambda (f(x)-A)-\mu_x (f(x)-0)\right ]dx$$
The optimization conditions are:
$$\begin{align}&\phi(f(x)+g(x))^{\phi-1}-\lambda=\mu_x\\ &\mu_xf(x)=0\\ &\mu_x\geq 0, f(x)\geq 0\\ &\int_0^1f(x)dx=A \end{align}$$
I honestly, don't really know where to start solving a problem with inequality constraint like this. I can draw the picture of what I think it should be, but I'm not sure how to do it analytically (without hand waving to the picture).