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Setting

Let $\Omega\subset\mathbb{R}^2$ be open and bounded and for $x,y\in\bar\Omega$ denote by \begin{equation*} C^{x,y} := \{\sigma\in W^{1,\infty}([0,1], \bar\Omega): \sigma(0) = x, \sigma(1) = y\}\,. \end{equation*} For $\varphi$ mapping $\bar\Omega$ to $\mathbb{R}_+$ define \begin{align*} L_\varphi(\sigma)&:= \int_0^1 \varphi(\sigma(t))|\dot\sigma(t)| \mathrm{d}t\\ c_\varphi(x,y) &:= \inf\{ L_\varphi(\sigma) \,|\, \sigma\in C^{x,y}\}\,. \end{align*}

Question

Which regularity assumptions on $\varphi$ suffice for $L_\varphi$ and $c_\varphi$ to be well defined? If $\varphi$ is less than lower semi-continuous, does the integral in the definition of $L_\varphi$ still exist?

Context

In this paper, the authors define $L_\varphi$ for continuous functions $\varphi$. After Proposition 3.1, they present a computation which uses $\varphi\in L^p$ and state that the computation is purely formal due to $L_\varphi$ and $c_\varphi$ not being well-defined for $\varphi\in L^p$. They then spend the elaborate paragraph §3.2 on extending both $L_\varphi$ and $c_\varphi$ to $L^p$. It's not clear to me why this is necessary.

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1 Answer 1

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I see the problem now: The set $\sigma([0,1])$ may be a null set and $\varphi\in L^p$ is only defined almost everywhere such that the integrand $\varphi(\sigma(t))$ is not well-defined in general.

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