Elliptic regularity when the Lagrangian is possibly infinite

I want to solve variational problems of the form $$\inf_u \int_{-1}^1 \phi(u'(x)) \text{ with } u(-1)=u(1) = 0,$$ where $$\phi(p)$$ is convex and is allowed to take on the value $$+\infty$$ for some values of $$p$$.

I would like to know about uniqueness, regularity, and so on, for these types of problems. Indeed, I am interested in parametric problems. Here is a toy "central path": $$u^*(x;t) = \arg\min_u \int_{-1}^1 tu + \phi(u') \text{ where } \phi(p) = -\log(1-p^2).$$ There's a physical interpretation to this, and an exact solution can be found. Despite the "badness" of $$\phi$$, it turns out that $$u^*(x;t)$$ is extremely smooth and converges very nicely (as $$t \to \infty$$) to the solution $$u(x) = |x|-1$$.

I'm looking at Gilbarg and Trudinger and I'm not finding it! Help!

Consider the 1d problem of minimizing $$\int_{-1}^1 g(x) u(x) + \phi(u').$$ All of the problems in the question can be reformulated in this form. Denote by $$p = u'$$, and note that $$u$$ can be recovered from $$p$$ by integration, using the boundary conditions (either from $$u(-1)=0$$ or $$u(1)=0$$). The strong form is $$(\phi'(p(x)))' = g(x) \text{ on } (-1,1).$$ Here, it is simplest to assume that $$\phi(p)$$ and $$g(x)$$ are even functions (which is the case of examples in the original question), in which case we can somehow integrate and also reduce the domain, to arrive at: $$\phi'(p) = \int_0^x g(t) \, dt =: G(x) \text{ for } x \in (0,1).$$ Assuming $$\phi$$ is strictly convex, then $$\phi'$$ is invertible and $$p(x) = \phi'^{-1}(G(x))$$ which is an "explicit" formula for the solution. In particular, if $$G(x)$$ is almost everywhere finite, then $$p(x)$$ is in the domain where $$\phi$$ is finite. Since the question is focused on the case where $$\phi$$ does indeed take on infinite values, if the domain of $$\phi$$ is a bounded interval, this proves that $$p$$ is bounded, and then $$u$$ is in $$W^{1,\infty}$$, which is already some sort of regularity. Furthermore, if $$\phi'^{-1}$$ and $$G$$ are both differentiable, then by the chain rule, so is $$p$$ and then $$u$$ is twice differentiable, etc...
We can see from the above that the case where the domain of $$\phi$$ is bounded is somehow better than the usual case where the domain of $$\phi$$ is $$\mathbb{R}$$, I am very much wondering whether that is the case in higher dimensions. I would like to believe that it is, but I can't find this stuff in the literature.