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cs89
  • Member for 10 years, 7 months
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revised
$C^{2}$ estimates for elliptic equations
use $ for latex in title
suggested
Approve
$C^{2}$ estimates for elliptic equations
awarded
 Fanatic
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Can a cubic polynomial in two real variables have three saddle points?
There is a somehow related question on Math.SE on whether a cubic polynomial in two variables can have exactly three critical points (and not four). The example I gave there has a degenerate critical point at $(0,0)$.
reviewed
Looks OK
Resolution of the diagonal for projective hypersurface
answered
Can we interpret fractional Sobolev spaces in terms of fractional derivatives?
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Sobolev estimates on domain with boundary
Cross-post from MSE: math.stackexchange.com/questions/4695311
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On Designing Some Optimal Control Problems
Could you give a concrete example of such a case?
awarded
 Explainer
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Is the product of $u \in W^{\sigma,1}(\Omega)$ and $v \in C^{0,\sigma}(\Omega)$ again in $W^{\sigma,1}(\Omega)$?
@shuhalo So the situation seems settled: as claimed in the book, the pointwise product does not work for the parameters of your question.
revised
Is the product of $u \in W^{\sigma,1}(\Omega)$ and $v \in C^{0,\sigma}(\Omega)$ again in $W^{\sigma,1}(\Omega)$?
deleted 247 characters in body
revised
Is the product of $u \in W^{\sigma,1}(\Omega)$ and $v \in C^{0,\sigma}(\Omega)$ again in $W^{\sigma,1}(\Omega)$?
fixed typo in seminorm def
comment
Is the product of $u \in W^{\sigma,1}(\Omega)$ and $v \in C^{0,\sigma}(\Omega)$ again in $W^{\sigma,1}(\Omega)$?
Condition (11) states that when you want $B^{s_1}_{p_1,q_1} \times B^{s_2}_{p_2,q_2} \hookrightarrow B^s_{p,q}$ with $s = s_1 = s_2$, you need $q \geq q_1$ and $q \geq q_2$.
suggested
Approve
Is the product of $u \in W^{\sigma,1}(\Omega)$ and $v \in C^{0,\sigma}(\Omega)$ again in $W^{\sigma,1}(\Omega)$?
answered
Is the product of $u \in W^{\sigma,1}(\Omega)$ and $v \in C^{0,\sigma}(\Omega)$ again in $W^{\sigma,1}(\Omega)$?
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Sobolev inequality with holes
There is also a recent rewriting Alberto Fiorenza, Maria Rosaria Formica, Tomáš G. Roskovec, and Filip Soudský. "Detailed proof of classical Gagliardo–Nirenberg interpolation inequality with historical remarks" Zeitschrift für Analysis und ihre Anwendungen 40, no. 2 (2021): 217-236 which is very helpful.
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Sobolev inequality with holes
@giorgio-metafune I believe he refers to Lecture II of Louis Nirenberg. "On elliptic partial differential equations." Annali della Scuola Normale Superiore di Pisa-Scienze Fisiche e Matematiche 13, no. 2 (1959): 115-162.
answered
Sobolev inequality with holes
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The compact embedding of $H^{1/2}_{2\pi}$ in $L^s(0, 2\pi)$
Sure, no problem. The other reference is indeed better suited for your question.
awarded
 Yearling
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