Well the way I was thinking was that the higher-income tenants would subsidise the lower-income ones -- but within a certain range (i.e., no more than 10% higher for the high-income ones, and 10% lower for the low income ones -- adjust as necessary). This could be implemented via something like a sales tax, but would apply to the monthly rent, and would be progressive based on the tenant's income (above a certain income level, tenants would pay more, and below a certain level tenants would receive a subsidy). The apartment owners would still get the same overall income though.

This should only have the effect of driving higher earners out of the city, which would then cause the desired effect of lowering property values, and therefore lowering rent overall.