| Here's the general rule -- you are generally entitled to use your own name as your business name, even if there is another company or entity that has trademark rights in a mark that includes your name. There are a few exceptions -- if the existing mark is what is known as a "famous mark," and it can be shown that your use of your name is intended, in bad faith, to divert customers to your site because of the famousness of the mark, or to disparage the existing mark, or to create something that you intend to sell to the owner of the famous mark -- then its possible to find infringement, even if you are using your own name.
But that's a pretty narrow exception -- the broad rule is that one is free to use one's own name in business, including a domain name.
All that said, this all comes down to what you want to do. From what you have written, it seems as though the owner of the famous mark would have little to no leverage against you -- you are using your own name, in a business entirely unrelated to their business. That said, the first step they will likely take -- an action to seize your domain name through the UDRP -- can be unpredictable. But typically, to seize a domain name, it requires a showing of a right to a mark and bad faith intent to profit from the similarity to the mark, which doesn't appear to be the case here.
Ideally, you could sit down with a local attorney, who could review the letter and all of the facts of your situation and advise you accordingly. If you have contact information, you could send them a letter, explaining your business and that you are using your own name, and that under 15 U.S.C. 1125(d) you lack the bad faith intent to profit from the fashion designer's mark, and that you are a bonafide user of the domain name for your own business, something along that line. You may want to read up on the code section just mentioned, along with the UDRP. | 
09-10-2009, 08:56 PM
Linear Mode
