Diff. Geom, Math. Phys., PDE Seminar: Hassan Jaber

• Date: 01/21/2014
• Time: 15:30
Lecturer(s):
Hassan Jaber (Université Henri Poincaré, Nancy 1)
Location:

University of British Columbia

Topic:

Hardy-Sobolev equations and related inequalities on compact Riemannian manifolds

Description:
Let (M,g) be a compact Riemannian Manifold without boundry of dimension n \geq 3, x_0 \in M, and s \in (0,2). We let \crit: = \frac{2(n-s)}{n-2} be the critical Hardy-Sobolev exponent. I investigate the influence of geometry on the existence of positive distributional solutions u \in C^0(M) for the critical equation

\Delta_g u+a(x) u=\frac{u^{\crit-1}}{d_g(x,x_0)^s}  \;\; \hbox{ in} \ M.

Via a minimization method, I prove existence in dimension n\geq 4 when the potential a is sufficiently below the scalar curvature at x_0. In dimension n=3, using a global argument, i prove existence when the mass of the linear operator \Delta_g + a is positive at x_0. On the other hand, by using a Blow-up around x_0, i prove that the sharp constant of the related Hardy-Sobolev inequalities on (M,g), which is equal to the one of the Euclidean Hardy-Sobolev inequalities, is achieved for all compact Riemannian Manifold of dimension n \geq 3 with or without boundary.
Other Information:

Location: ESB 2012
{"ft_lang_label":"__label__en","ft_lang_prob":0.7465975,"math_prob":0.99207664,"size":1928,"snap":"2019-43-2019-47","text_gpt3_token_len":559,"char_repetition_ratio":0.10966736,"word_repetition_ratio":0.67333335,"special_character_ratio":0.2650415,"punctuation_ratio":0.13417721,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99436766,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-16T14:01:25Z\",\"WARC-Record-ID\":\"<urn:uuid:6a0da293-d0c7-432b-945c-5da0593edde0>\",\"Content-Length\":\"17772\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:fae77e6f-2296-4072-8b92-7f03b78ab2ea>\",\"WARC-Concurrent-To\":\"<urn:uuid:ab3f14c3-d1dc-42d1-bf86-ed5029c80916>\",\"WARC-IP-Address\":\"137.82.151.209\",\"WARC-Target-URI\":\"https://www.pims.math.ca/scientific-event/140121-dgmppshj\",\"WARC-Payload-Digest\":\"sha1:XBBA5FR66F5NA7SYBCUALS6R7VHECEHJ\",\"WARC-Block-Digest\":\"sha1:IHQ36N5PSWSKPTIT6AFALKDUNDQIHSYF\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986668994.39_warc_CC-MAIN-20191016135759-20191016163259-00063.warc.gz\"}"}