Keep oven preheated at 390 F.
In a large bowl, whisk the cream cheese.
Add the sugar in two additions as you beat the cheese for about 2 minutes.
Add corn flour, all purpose flour, vanilla extract, lemon juice and yogurt, one by one, whisking in between.
Whisk for about 30 seconds to 1 minute.
Add the milk and mix until you see no runny milk
Divide the batter in the following way for each of the colors : 1 ½ cup - Red, 1 ¼ cup - Orange, 1 cup - Yellow, ¾ cup - Green, ½ cup - Blue and ¼ cup - Violet.
Pour the batter into the previously prepared pan, in the following order: Red, orange, yellow, green, blue and violet, pouring very carefully in the center.
Bake at 390 F for 20 minutes. Then reduce the temperature to 230 F and bake for another hour.
Switch the oven off. Keeping the oven door slightly open, let the cake cool inside the oven for 1 hour.
Now place the pan on a wire rack and let it cool at room temperature.
Cover loosely with a foil and refrigerate it overnight or about 6 to 8 hours.
After an overnight cooling, release the cake from the pan using a sharp knife.
Slice and serve! (To see how to cut a clean slice, check out the heading " How to Make Perfect Rainbow Cheesecake" )