should header be fixed when using three column layout?

I'm designing three column layout website where all columns scroll independently. There is no problem on large displays, if header(I have large header) is fixed, but on smaller screens (1366x650 px) I have left small space for content.

If header is not fixed it performs strange when scrolling.

Are there any rules what to do with headers when using 3 column layouts? And if fix header what to do with small space for content on smaller screens? Thanks in advance