NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2015, 6 (1), P. 46–56
ON THE ROBIN EIGENVALUES OF THE LAPLACIAN IN THE EXTERIOR OF A CONVEX POLYGON
Konstantin Pankrashkin – Laboratoire de mathématiques, Université Paris-Sud Bâtiment 425, 91405 Orsay Cedex, France; konstantin.pankrashkin@math.u-psud.fr
Let Ω⊂ R2 be the exterior of a convex polygon whose side lengths are l1, . . . , lM. For a real constant α , let HαΩ denote the Laplacian in Ω, u → -Δu with the Robin boundary conditions ∂u/∂v = αu at ∂Ω, where v is the outer unit normal. We show that, for any fixed m∈N, the mth eigenvalue EmΩ(α) of HαΩ behaves as EmΩ(α) = -α2 + μmD + O(α-1/2) as α → +∞ where μmD stands for the mth eigenvalue of the operator D1 ⊕ . . . ⊕ DM and Dn denotes the one-dimensional Laplacian f → -f” on (0,ln) with the Dirichlet boundary conditions.
Keywords: eigenvalue asymptotics, Laplacian, Robin boundary condition, Dirichlet boundary condition.
PACS 41.20.Cv, 02.30.Jr, 02.30.Tb
DOI 10.17586/2220-8054-2015-6-1-46-56
I am really impressed with your writing skills and also with the layout on your blog. Is this a paid theme or did you customize it yourself? Anyway keep up the excellent quality writing, it is rare to see a nice blog like this one today..
It is in reality a great and helpful piece of information. I am satisfied that you just shared this useful info with us. Please keep us informed like this. Thanks for sharing.
I in addition to my guys happened to be viewing the excellent pointers from your web site and so quickly got an awful suspicion I had not expressed respect to the web site owner for those secrets. My boys ended up for that reason glad to read them and now have quite simply been tapping into those things. Appreciate your being well thoughtful as well as for deciding upon this kind of useful subjects millions of individuals are really needing to understand about. My sincere apologies for not saying thanks to you earlier.
This design is incredible! You certainly know how to keep a reader entertained. Between your wit and your videos, I was almost moved to start my own blog (well, almost…HaHa!) Wonderful job. I really loved what you had to say, and more than that, how you presented it. Too cool!
I always was concerned in this topic and stock still am, thankyou for posting.
It’s a pity you don’t have a donate button! I’d most certainly donate to this brilliant blog! I suppose for now i’ll settle for bookmarking and adding your RSS feed to my Google account. I look forward to new updates and will share this website with my Facebook group. Talk soon!
I was curious if you ever considered changing the structure of your site? Its very well written; I love what youve got to say. But maybe you could a little more in the way of content so people could connect with it better. Youve got an awful lot of text for only having 1 or 2 images. Maybe you could space it out better?
Amazing! This blog looks just like my old one! It’s on a completely different subject but it has pretty much the same page layout and design. Outstanding choice of colors!
Someone essentially help to make severely posts I would state. That is the first time I frequented your web page and up to now? I amazed with the analysis you made to make this actual post amazing. Wonderful task!
I see something truly special in this internet site.
Some truly interesting info , well written and loosely user genial.
I cling on to listening to the news update lecture about receiving free online grant applications so I have been looking around for the finest site to get one. Could you advise me please, where could i get some?
You have mentioned very interesting details! ps nice web site.
I likewise think thence, perfectly written post! .
Does your website have a contact page? I’m having a tough time locating it but, I’d like to shoot you an email. I’ve got some recommendations for your blog you might be interested in hearing. Either way, great site and I look forward to seeing it grow over time.
You have observed very interesting points! ps nice internet site.
Good write-up, I am regular visitor of one?¦s blog, maintain up the excellent operate, and It’s going to be a regular visitor for a long time.
Some truly great information, Sword lily I detected this. “The outer conditions of a person’s life will always be found to reflect their inner beliefs.” by James Allen.
I have been exploring for a little for any high-quality articles or blog posts in this sort of space . Exploring in Yahoo I ultimately stumbled upon this site. Studying this information So i’m glad to exhibit that I have a very excellent uncanny feeling I found out exactly what I needed. I most indubitably will make sure to don’t omit this site and provides it a glance on a continuing basis.
I was wondering if you ever considered changing the structure of your site? Its very well written; I love what youve got to say. But maybe you could a little more in the way of content so people could connect with it better. Youve got an awful lot of text for only having 1 or 2 pictures. Maybe you could space it out better?
Heya i’m for the first time here. I came across this board and I find It truly useful & it helped me out much. I hope to give something back and aid others like you aided me.
When I originally commented I clicked the “Notify me when new comments are added” checkbox and now each time a comment is added I get three e-mails with the same comment. Is there any way you can remove me from that service? Bless you!
I’ll immediately grasp your rss as I can’t in finding your email subscription link or newsletter service. Do you’ve any? Kindly let me realize in order that I may just subscribe. Thanks.
I was just searching for this info for a while. After six hours of continuous Googleing, finally I got it in your web site. I wonder what is the lack of Google strategy that don’t rank this type of informative sites in top of the list. Usually the top sites are full of garbage.
Thank you for the good writeup. It in fact was a amusement account it. Look advanced to far added agreeable from you! However, how can we communicate?
I have been surfing online more than three hours today, but I never found any interesting article like yours. It?¦s beautiful value sufficient for me. In my opinion, if all webmasters and bloggers made just right content material as you did, the internet will likely be much more helpful than ever before.
I like what you guys are up too. Such clever work and reporting! Keep up the excellent works guys I’ve incorporated you guys to my blogroll. I think it will improve the value of my web site
.
Great line up. We will be linking to this great article on our site. Keep up the good writing.
F*ckin’ remarkable issues here. I’m very happy to peer your post. Thanks so much and i’m looking forward to contact you. Will you kindly drop me a mail?
Utterly indited articles, thank you for selective information. “The last time I saw him he was walking down Lover’s Lane holding his own hand.” by Fred Allen.
I have been examinating out some of your articles and i can state pretty good stuff. I will definitely bookmark your blog.
I do trust all the concepts you have offered to your post. They are very convincing and can definitely work. Nonetheless, the posts are too quick for newbies. May you please lengthen them a little from next time? Thank you for the post.
Blue sant trick for men
Merely wanna say that this is extremely helpful, Thanks for taking your time to write this.
Very interesting info !Perfect just what I was searching for! “…obstacles do not exist to be surrendered to, but only to be broken.” by Adolf Hitler.
Utterly pent subject matter, regards for entropy.
I am glad to be one of many visitants on this great site (:, thanks for putting up.
Very interesting details you have mentioned, appreciate it for putting up. “The best time to do a thing is when it can be done.” by William Pickens.
Thanks a lot for sharing this with all people you really recognize what you are talking approximately! Bookmarked. Kindly additionally consult with my site =). We can have a hyperlink trade contract among us!
I dugg some of you post as I thought they were handy handy
I was suggested this website by my cousin. I’m not sure whether this post is written by him as nobody else know such detailed about my problem. You are wonderful! Thanks!
Fantastic blog you have here but I was wanting to know if you knew of any message boards that cover the same topics discussed in this article? I’d really like to be a part of group where I can get opinions from other experienced people that share the same interest. If you have any recommendations, please let me know. Bless you!