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Dear community.

I'm looking for survey articles about recent and old results about stability for the elliptic and backward-parabolic Cauchy problem.

For example: Take $\partial_t u + a(t)\partial^2_x u = 0$ with $u(0,x)=u_0(x)$. What is the minimal regularity-assumption on $a$, in terms of a first or second order modulus of continuity to get Hölder continuous dependence of the solution on the initial data. Or: What is the best condition like $|f(t)a(t)|\leq C$ on $a\in C^1(0,T]$ to get the same result. Another question is what happens if we mix this two conditions.

Often is claimed, that Lipschitz is the optimal regularity. But I think this is, like for uniqueness, not true. At least for Zygmund there should be a similar result and also maybe for coefficients with a first order modulus of continuity $\mu$ satisfying the Osgood Condition: $\int\limits_0^1 \frac{ds}{\mu(s)}=+\infty$.

Maybe there is not so much in this direction. For hyperbolic equations there are a lot of results with this kind of conditions.

Best CJ

edited Apr 10 2011 at 12:09

asked Apr 9 2011 at 14:02

CJ
93●4

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Why? What is your background? And what have you searched in? The first hit in Google for "elliptic cauchy stability" already seems relevant: iopscience.iop.org/0266-5611/25/12/123004 – Willie Wong Apr 9 2011 at 18:33

@Wong: I'm trying something in this direction and I just want an overview about results and methods used to prove such kind of results. Especially I'm interested in the minimal regularity of the coefficients. I searched with google and found this and of course a lot of original papers devoted to the problem. But beside the Alessandrini ed al paper I couldn't really find good reviews on the topic. – CJ Apr 10 2011 at 11:55

You are aware of course that a backwards parabolic problem can not be solved unless you put extremely strong conditions on the data (i.e. of Gevrey class of order less than 1/2), and when you are in such a situation the regularity of the coefficient does not really matter i.e. a locally integrable $a(t)$ is fine? – Piero D'Ancona Apr 10 2011 at 12:47

@D'Ancona: Yes, I'm aware of the fact that this equation, as the elliptic, is ill-posed even in reasonable situations. I'm just looking for uniqueness and stability (in the sense of John) at the moment. – CJ Apr 10 2011 at 16:00

@CJ: what is "stability in the sense of John"? Can you give a (hopefully easy to find online) reference? – drbobmeister Apr 11 2011 at 0:48

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References? Stability of the Cauchy problem for elliptic and backward-parabolic operators.

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