It would seem practical to me for you to investigate the source of the sinkhole before trying to find out how to cure the problem.

Is it from damaged field tile?  Is it from mine subsidence?  Is it from natural underground features, like a cave under your land?  Try to do some research to find out what is causing the problem, and you will be closer to your solution.

My first thought is to check with my local NRCS (Used to be Soil Conservation Service) , if they don't have answers, they should be able to suggest authorities such as the USGS (United States Geological Service) to help you do your investigation and possibly offer solutions.