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 m in a particular class of continuous functions, that there is a variational integral and boundary conditions such that, for every p from [1, infinity], the infimum is equal to m(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