Essayez de le changer en ceci ;

   location / {
       root   /home/mywebsite/public;
       index index.php index.html index.htm;
       try_files $uri $uri/ /index.php;

   }

http://wiki.nginx.org/HttpCoreModule#try_files

Il fonctionne parfaitement.

Et pour activer PHP

passer les scripts PHP au serveur FastCGI écoutant sur 127.0.0.1:9000

    #
    location ~ \.php$ {
            fastcgi_split_path_info ^(.+\.php)(/.+)$;
    #       # NOTE: You should have "cgi.fix_pathinfo = 0;" in php.ini
    #
    #       # With php5-cgi alone:
    #       fastcgi_pass 127.0.0.1:9000;
    #       # With php5-fpm:
            fastcgi_pass unix:/var/run/php5-fpm.sock;
            fastcgi_index index.php;
            include fastcgi_params;
    }