Reference request: functional analysis results used in Taubes paper (1980)

Question

I'm studying Taubes paper 'Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations'. I'm looking for a reference of three following theorems:

1. Let $f(x)$ be a convex funtional defined in a open convex set of a normed space $E$. Let $f(x)$ be a real Gateaux differentiable functional with $f'(x,\cdot )$ continouos for fixed $x\in E$. Then $f(x)$ is weakly lower semi-continuous.

2. If a strictly convex functional $f(x)$ defined in a linear space $E$ has a minimum at a point $x_0$, then $x_0$ is an absolute minimum point, and there are no other minimum poins.

3. Let $f(x)$ be a real Gateaux differentiable funtional defined in a real reflexive Banach space $E$, which is weakly lower semi-continuous and satisfies the condition $f'(x,x)>0$ for any vector $x$ in $E$, $|x|=R>0$ and $f'(x)=gradf(x)$. Then there is exists an interior point $x_0$ of the ball $|x|\leq R$ at which $f(x)$ has a local minimum so that $f'(x)=0$.

Taubes paper cites the book 'Variational method and method of monotone operators in the theory of nonlinear equations' by M. M. Vainberg. However, the library in my university does not have this book. Is there another reference where I can find these results?

Thanks.

Answer 1

2 and 3 are easy enough:

2. If $y$ is a point with $f(y)\leq f(x_0)$ then by (strict!) convexity, for all $t\in [0,1]$, $$ f((1-t)x_0+ty)< (1-t)f(x_0)+tf(y)\leq f(x_0). $$ Taking $t\to0$ yields a contradiction to the fact that $f$ has a minimum at $x_0$. Therefore the minimum is global and unique.

3. If $f$ had a minimum at $x$ with $|x|=R$, then for $h$ sufficiently small $f(x-hx)-f(x)\geq 0$. By the definition of the Gateaux derivative this means $-f'(x,x)\geq 0$, so your condition means no minimum is at the boundary. On the other hand since $E$ is reflexive, the closed balls are compact in the weak topology. Take a minimizing sequence $x_k$ with $f(x_k)\to a:=\inf_{|y|\leq R} f(y)$, then by compactness $x_k$ has a convergent subsequence (say to $x$, with $|x|\leq R$) and by lower semicontinuity we have $f(x)\leq \liminf_k f(x_k)= a$. By the previous argument, we must have $|x|<R$ and we're done.

Edit: As for a reference, I think Jost's "Calculus of Variations" should have this (except maybe 1).