If you don't suppress errors when sending the email do you get an error?
Have you verified you get a value for "trim($r_chk[0]['email'])"?
Also try sending just a basic email. ie..
mail('you@domain.com', 'test', 'test message', 'From: doesnt@matter.com');
Check to see if that goes through, if it doesn't then you have a configuration problem with sendmail on the server. If it does then its most likely an issue with the headers, although I don't see anything wrong with them offhand