Avoiding notice period

Hi, I'm a software engineer, working for cognizant technology solutions, Bangalore from past two years. I joined this company as a fresher, now that I have got a better offer in Mindtree, Bangalore. I want to switch my job. But the problem is of notice period, cognizant has a policy of 2 months notice period, they even have buyout option. I submitted my resignation on 21st October 2015, so I'm still on notice period, according to my managers they can relive me of duty only after 16th December 2015, which is fair according to company rule. But I got my joining day to mindtree on 7th December 2015. 1. I spoke to mindtree HR to postpone my joining post 16th December, but since its year end and they don't have joinings after December 7th, they are helpless in offering any solution to me 2. I tried to persuade my managers and HR in cognizant for an early release, but it's of no use. And in the process I realised how evil are HRs and Managers, they are not at all ready to listen to my problems not even ready to give me buyout option instead my manager spoke to me in a very harsh way. And dismissed my pleas in humanly, As always HR is just a supporter to managers they never bother about employee problems. After this my manager started to torture me with humongous work loads every day. Back at home my dad fell sick recently, now my salary is the only source of income to our family, and this job is very important to me, it's impossible to get this kind of pay hike from other companies and it'll help my family a lot, I'm losing psychological balance due to this unable to concentrate on work. Please suggest any ways to get out of this situation and join mindtree on date. P. S. Please consider suggesting negative ways too, such as a lie or fake documentation etc.. Bcoz there is no point in being honest with a company that doesn't care of employee's problems. Thanks, Ranjan Manohar