Thank you till,
I tested the sending of mails with Outlook Express and Thunderbird. In Outlook I checked the opton "SMTP server needs authentication". On Thunderbird I use "Use username and password". But the problem is still the same.
The domain is already in /etc/postfix/local-host-names.
It's very strange?! Any other ideas? I searched the forum already but I still haven't found anything like that.
Perhaps it's easier when I show my /etc/postfix/main.cf:
mydomain = anotherdomain.de
myhostname = server1.$mydomain
smtpd_sasl_auth_enable = yes
smtpd_sasl_security_options = noanonymous
broken_sasl_auth_clients = yes
smtpd_recipient_restrictions = permit_sasl_authenticated,permit_mynetworks,reject_unauth_destination
inet_interfaces = all
alias_maps = hash:/etc/aliases
smtpd_tls_auth_only = no
smtp_use_tls = yes
smtpd_use_tls = yes
smtp_tls_note_starttls_offer = yes
smtpd_tls_key_file = /etc/postfix/ssl/smtpd.key
smtpd_tls_cert_file = /etc/postfix/ssl/smtpd.crt
smtpd_tls_CAfile = /etc/postfix/ssl/cacert.pem
smtpd_tls_loglevel = 1
smtpd_tls_received_header = yes
smtpd_tls_session_cache_timeout = 3600s
tls_random_source = dev:/dev/urandom