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Results tagged with sobolev-spaces
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A Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order.
12
votes
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History of Sobolev space notations
The history of both the name "Sobolev space" and the notation (which changed over the years), has been well described by J. Naumann [link to pdf file]. The history of the name is particularly amusing: …
- 188k
0
votes
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Definition of Euler-Lagrange equation and properties, where can I find?
These lecture notes by Piotr Hajłasz might have the introductory level you are looking for:
The lectures will be divided into two almost independent streams. One
of them is the theory of Sobolev spac …
- 188k
9
votes
Accepted
Prove J.L. Lions’s Lemma without using Fourier transform
Lion's lemma is equivalent to several other properties that have simpler proofs, see On a lemma of Jacques-Louis Lions and its relation to other fundamental results:
Some of these equivalent properti …
- 188k
4
votes
Euler-Lagrange equations for minimizer of energy with indicator function
Edit: An earlier version of this answer missed a factor of two, now corrected, thanks to Daniele Tampieri.
We seek the variation of the functional
$$L[u]=\int_\Omega\left(|\nabla u|^2+1\right)\chi_{u> …
- 188k
13
votes
Accepted
Caccioppoli-Leray Inequality for De Giorgi's theorem proof
I made a trip to the library and scanned the relevant pages from Miranda's 1955 book:
page 152-153 and page 154-155
the references are:
[3] J. Leray, J.Math. pures et appl. 17, 89-104 (1938)
[8] R. Ca …
- 188k