Timeline for Maximizing $\|u\|_q\|u\|_p$ over the unitary sphere in the Sobolev space $H_0^1(\Omega)$
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Aug 7, 2016 at 22:15 | comment | added | Pietro Majer | Exact (more precisely, I guess I should have said "u has constant sign", so u can be assumed nonnegative w.l.o.g.) | |
| Aug 7, 2016 at 17:15 | vote | accept | Tomás | ||
| Aug 7, 2016 at 17:13 | comment | added | Tomás | I see. If $u$ is a maximizer, we can use the Lagrange multiplier theorem to infer that $-\Delta u=\lambda |u|^{p-2}u$. Then, an argument with the test function $v=u^-$ yields that $u\ge 0$. The rest is as you said. Thank you. | |
| Aug 7, 2016 at 12:31 | history | edited | Pietro Majer | CC BY-SA 3.0 |
added 12 characters in body
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| Aug 7, 2016 at 8:48 | history | answered | Pietro Majer | CC BY-SA 3.0 |