Let $a,b \in \mathbb R$ and consider the functional $J$ on $X$:


$$J[u] = \int\_0^1 \left( (u'(x))^2 -a)^2 + b \ln (1+ u^2(x))\right) dx$$


Providing reasons specify if the $\inf J$ over $X$ is attained of not


1. $a>0, b \geq 0, X =\{ u \in C^1([0,1]); u(0)=0, u(1)=0 \}$
2. $a>0, b > 0, X =\{ u \in AC([0,1]); u(0)=0, u(1)=0 \}$
3. $a<0, b \geq 0, X =\{ u \in AC([0,1]); u(0)=0, u(1)=1 \}$


**My proof**


I am trying to apply the following theorem to find the answer.


**Special version of Tonelli’s theorem**


Assume that the function $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{R}$, $g(x, \xi): [a,b] \times \mathbb{R} \to \mathbb{R}$ are continuous, $f$ is bounded below, $g$ is convex in $\xi$ and satisfies


$$\exists r>1,\, \exists C>0\,\, \text{such that}\,\, g(x,\xi) \ge C| \xi|^r,\,\, \forall (x, \xi) \in [a,b] \times \mathbb{R}.$$


Then there exists a minimizer of the functional
$$J[u] = \displaystyle\int\_a^b (f(x,u(x)) + g(x,u'(x))) dx$$ in the space $X= \{ u \in AC([a,b]); u(a)=\alpha, u(b)= \beta \}.$


Here I considered $f(x,u(x)) = b \ln (1+u^2(x))$ and $g(x,u'(x))= (u'(x)^2 -a)^2$. Am I in the right path? I am not sure on what should I do next.



Which set of functions admits the existence of the minimizer?
The answer to (3) is yes. Indeed, then all the conditions of what you call "special version of Tonelli’s theorem" (proved in [this answer](https://mathoverflow.net/a/413645/36721)) are satisfied.


The answer to (2) is no. Indeed, for natural $n$ and $x\in[0,1]$, let
$$u\_n(x):=\sqrt a\,d(x,E\_n),$$
where $E\_n$ denotes the set $\{0,\frac1n,\frac2n,\dots,\frac nn\}$ and $d(x,E\_n)$ denotes the shortest distance from $x$ to the set $E\_n$. Then $u\_n\in AC[0,1]$ and
$$J[u\_n]=b\int\_0^1\ln(1+u\_n^2(x))\, dx\le b\ln(1+a/(4n^2))\to0$$
as $n\to\infty$. Since $J[\cdot]\ge0$, it follows that $\inf J[\cdot]=0$. However, this zero infimum is not attained at any $u$ -- if $J[u]=0$, then $u=0$ and hence $J[u]=a^2>0$.


The answer to (1) is no as well. This follows because the $u\_n$'s as above can be appropriately approximated by $C\_1$ functions.


