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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!

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Strictly speaking, my question is about the 1d case but I'm really interested in the 2d case so I'm leaving it open. However, I'm giving here a (partial) answer for the 1d case.

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.

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