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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.
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
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
0
votes
Accepted
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
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
12
votes
Accepted
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