Page works with SSL, but not internally
Try this:

$base_protocol = empty($_SERVER['HTTPS']) || strtolower($_SERVER['HTTPS']) != 'on' ? 'http' : 'https';
$base_url = $base_protocol . '://www.mysite.com';


Though you may need the Secure Pages module for it to work, or to have redirection set up in your .htaccess file or virtualhost.