myna.me for a web app in a niche market is just fine, especially if you think it works well as a brand for your marketing.
Web designers should be more aware that the
.me ccTLD is often used for personalized websites. Many large corporate sites (e.g., Facebook, Google, Visa, etc...) have also registered
.me domains for URL shortening purposes too. See this for an example list.
If the owner of the
myname.com isn't in the same industry and is already operating a website, it should be less concerning as to whether they'd change their site to compete with yours, or do this in attempt to redirect traffic from your website (should it become that popular outside your niche).
As the other answer points out, if there is a trademark for
myname in your country, that might be more of a concern; However if they are in a different industry, it would be difficult for them to assert that you did not have a legitimate interest in the domain or registered it in "bad faith", which is two out of three elements needed to succeed in the UDRP domain dispute process.
In short, if you're concerned with competition, I'd suggest looking to see if other domain extensions were taken and what websites they contained to gauge if they were similar and might potentially compete with your site. Then do a trademark search in your local country for
myname to make sure that it isn't protected there. By the way, something as generic as
myname would be difficult to trademark because it's commonly used, as apposed to something more specific such as
Lastly, it's always a good idea to register your trademark as soon as possible to establish the priority right for your mark, and to avoid potential domain disputes down the line with those who might register it.