Examples of the Lavrentiev Phenomenon with Continuous Sobolev Exponent Dependence

We construct variational problems with infima that have non-trivial continuous dependence upon the Sobolev space from which the competing functions are taken. It is shown, for each $\mu$ in a particular class of continuous functions, that there is a variational integral and boundary conditions such that, for every p from $[1,\infty]$, the infimum is equal to $\mu(p)$ if the admissible class is a subset of W^{1, p}. Thus, the manner in which the infimum depends upon the Sobolev exponent may be prescribed