I ended up using (note the '.log' filename and the single quotes around 'myfilename_'):
<rollingStyle value="Date" />
<datePattern value="'myfilename_'yyyy-MM-dd"/>
<preserveLogFileNameExtension value="true" />
<staticLogFileName value="false" />
<file type="log4net.Util.PatternString" value="c:\\Logs\\.log" />
This gives me:
myfilename_2015-09-22.log
myfilename_2015-09-23.log
.
.