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Results tagged with ap.analysis-of-pdes
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user 12898
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
2
votes
Accepted
question about mixed spectrum of a linear operator $\mathcal{L}$
Well, I have to confess that I am not sure about your goals. In which space are you working? But there is always a way to represent the solution as a generazied exponential function, called operator s …
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4
votes
Maximum principle for weak solutions
Yes. If you use operaor semigroups to represent the solutions, you can infer the positivity of the mild solutions (which are the same as the weak solutions) immediately.
There is an extensive treatm …
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4
votes
When does $\{u\in H^1_0: \Delta_{\mu}u\in L^2\}=H_0^1\cap H^2$.
Renardy-Rogers, Theorem 9.53 states that $C^2$ boundary is sufficient. Example 9.52 before shows that this cannot be heavily relaxed.
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1
vote
Ergodic Mean for Schrodinger flow
Ruess and Summers have a generalization of the geometric ideas you describe to nonlinear contraction semigroups. I suggest you take a look.
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3
votes
Reference Request: Schauder theory for fourth-order parabolic equations
I do not have access to the book you cite, but Section 3.2 in Lunardi's book contains a broad overview on higher order parabolic problems with lots of references, including the most important Hölder e …
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2
votes
Accepted
Easy question on Sobolev spaces
I would like to expand a bit what Delio said. Your question is a bit confusing as it is, but we may assume that you mean that $p$ and $q$ are the parameters representing the derivatives. Then what you …
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2
votes
Robin-Laplacian in unbounded domains
For the case $p=2$, I would have a look at this paper by Wolfgang Arend and Mahamadi Warma, and its follow-up papers: Potential Analysis 19: 341–363, 2003.
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6
votes
Accepted
Proof that $L^2(0,T;X)^* = L^2(0,T;X^*)$
To give you a reference: Diestel-Uhl, Vector measures, page 98, Chapter 4, Theorem 1:
$$L^p(\mu,X)^\ast = L^q(\mu,X^\ast)$$
if and only if $X^\ast$ has the Radon-Nikodym property with respect to $\m …
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1
vote
Reference for Neumann-Laplacian
Have a look at Lunardi's book, it is more on the functional analytic questions you have:
http://books.google.co.uk/books/about/Analytic_semigroups_and_optimal_regulari.html?id=mWojiHzg9bEC
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3
votes
Must Neuman Elliptic operator has discrete spectrum ?
Of course, on a general domain, the question os how do you define the Neuman Laplacian. There is an excellent exposition in
W. Arendt, A.F.M. ter Elst: Sectorial forms and degenerate differential op …
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7
votes
Reference request: parabolic PDE
If you are interested in sup-norm and Hölder estimates, then Lunardi's book is a good start:
http://www.amazon.com/Semigroups-Regularity-Parabolic-Differential-Applications/dp/3764351721/ref=la_B001K …
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4
votes
Real analysis on vector-valued spaces, $L^{p}(\mathbb{R}^N,E)$ ,$H^{s}(\mathbb{R}^N,E)$
You should brows the papers of Amann and his students for this type of results. You will find a lots of interesting results in the paper
http://user.math.uzh.ch/amann/files/cevvss.pdf
about embeddin …
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7
votes
Accepted
Abstract ODE; PDE; uniqueness of solution
No, this is not true. There is no backward uniqueness in general.
What you need is the theory of operator semigroups, and here is a simple example.
Consider the operator $Af=f'$ in the space $X=L …
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2
votes
Accepted
$C_0$ semigroups on parameterized Banach spaces or moving domains
You will not find many things about such operator families in the literature, but what there is is for the non-autonomous case. See for example
M. Geissert & A. Lunardi Invariant Measures and Maxima …
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3
votes
Accepted
PDEs on torus $\mathbb T$
It is a bit old, but I would check Bourgain first.
A more recent reference is (among many others) the preprint of Strunk.
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