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

in a particular class of continuous functions, that there is a variational integral and boundary conditions such that, for every p from

[1,\infty] [ 1 , ∞ ]

, the infimum is equal to

\mu(p) μ ( 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