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Results tagged with fa.functional-analysis
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user 22238
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
5
votes
Accepted
Weak continuity of Lebesgue decomposition
The answer is no. For example, you can construct a sequence of $g_n\in L^1(\mathbb{R}^n)$ converging to the Dirac $\delta$ measure.
Furthermore, we can also construct a sequence of singular measures …
- 190
0
votes
Defining surface integral on boundary of $C^1$-domain
A general way to define this is to use rectifiable sets. For example, if $\Omega$ is a set of finite perimeter, measurable functions and their integrations are well defined. Moreover, the divergence t …
- 190
0
votes
How to show this integral on boundary of Lipschitz domain is finite?
You can use the dyadic decomposition. The integral is controlled by
$$\sum_{k=0}^{+\infty}2^{k(n-2)}\mbox{Area}(\partial\Omega\cap (B_{2^{-k+1}}\setminus B_{2^{-k}})).$$
The next problem is to give an …
- 190
2
votes
Accepted
Checking initial data in parabolic PDE with no control on time derivative
The fundament solution approximates the $\delta$ function as $t\to0^+$. Thus if we represent the solution as an integral, it converges to the initial value as $t\to 0^+$ in $L^2$. This is similar to t …
- 190