Timeline for Second-order elliptic regularity with rough coefficients

8 events

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Dec 27, 2021 at 5:24	comment	added	Math604		I think i had something like $ -\Delta u(x) + b(x) u_{x_1} = f(u)$ and $b$ was not smooth enough across say a hyperplane like $x_1=0$ to apply the regularity theory to get $u$ was $C^{2,\alpha}$. But $u$ was an even reflection across $x_1=0$ and hence $u_{x_1}=0$ on the hyperplane. WHen you examined exactly the term $ b(x) u_{x_1}$ it was holder continuous and hence you could get what you wanted.
Dec 27, 2021 at 4:57	comment	added	atzol		@Math604 Thank you. Even though your example is quite specific, I would be interested to read it in more detail if you have time to write it as an answer.
Dec 27, 2021 at 4:56	vote	accept	atzol
Dec 25, 2021 at 22:52	comment	added	Math604		I had exactly something like this come up once. I had solutions on a cube that were reflections and the $b$ term didn't have enough regularity to get what I wanted. But when you viewed the product of $b$ and the first derivatives you could apply the Holder theory you wanted. But of course this was very special case.
Dec 24, 2021 at 15:45	answer	added	username		timeline score: 3
Dec 24, 2021 at 0:44	history	edited	YCor	CC BY-SA 4.0	removed capitals from title
S Dec 24, 2021 at 0:29	review	First questions
Dec 24, 2021 at 0:46
S Dec 24, 2021 at 0:29	history	asked	atzol	CC BY-SA 4.0