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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

7 votes
Accepted

How do people prove $\Gamma$-convergence in more complicated settings?

I am mostly familiar with the simpler definition ("Definition in first-countable spaces" from the Wikipedia link): Given the functionals $F_\varepsilon, F: X \to \overline{\Bbb{R}}$ (indexed for $\va …
Beni Bogosel's user avatar
  • 2,222
9 votes
4 answers
2k views

Books about capacity theory

While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for eve …
2 votes

Strange result about convexity

Here is a more problem solving approach, since this problem comes from AoPS. Instead of working with $f$, work with $g = f''$ which is a convex function. Note that it is enough to assume $g(1)=0$. In …
Beni Bogosel's user avatar
  • 2,222
5 votes
1 answer
507 views

$C_0$-semigroups applications

My graduation thesis was about stability theorems for $C_0$-semigroups (see the Wikipedia article for the definitions: http://en.wikipedia.org/wiki/C0-semigroup). I would like to know if there is som …
3 votes
2 answers
1k views

Sum of two closed operators closable

I found this question on another forum, and after processing it a bit, I didn't find a good answer. The question is: Is the sum of two closed operators closable? If not, give an example of two clo …
1 vote
0 answers
134 views

Inequality involving BV norm and a regularizing kernel

In the same article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/# (related to this ques …
1 vote

Are weak and strong convergence of sequences not equivalent?

A remark in H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Chapter 3: In any infinite dimensional space the weak topology is strictly coarser than the strong t …
Beni Bogosel's user avatar
  • 2,222
2 votes
0 answers
175 views

A limit involving a regularizing kernel

I am studying the following article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/# Some …