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Results tagged with sobolev-spaces
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user 51695
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.
1
vote
Accepted
Non convex optimization problem in $W_0^{1,2}$
You can treat this as a problem with two Lagrange multipliers. Then by standard methods, a minimizer $f$ has to exist (by convexity in $f'$) and has to be a weak solution to
$$-f'' + \lambda f + \mu f …
- 2,504
3
votes
Accepted
Must a Schauder basis for $W^{1,p}_0(\Omega)$ be oscillatory?
Even the modified question does not hold.
Let $u_n$ be a basis such that $\mathcal{L}^d(\operatorname{spt} u_n) \to 0$, e.g. a wavelet basis and let $\phi \in C_0^\infty(\Omega)$ a function such that …
- 2,504
2
votes
Accepted
Does weak continuity of Jacobians hold for non nondegenerate maps?
There is a counterexample, however there might be ways to avoid it.
Take $\mathcal{M} = \mathcal{N} =\mathbb{S}^2$, but now consider sequence of maps that cover the sphere twice, where you shrink the …
- 2,504
4
votes
Accepted
Averaging the mass of a Sobolev function $f\in W^{1,p}(\Omega)$ near $\partial\Omega$
In Evans & Gariepy's "Measure theory and fine properties of functions", Sec. 5.3., they construct the trace operator on a bounded Lipschitz domain $\Omega$ for BV-functions (and thus by inclusion for …
- 2,504
2
votes
Accepted
Is a locally invertible weak limit of injective maps injective almost everywhere?
Okay, let me try a writeup of the comment chain. For any reasonable subset $A\subset \Omega_2$ and $B := f^{-1}(A)$ you get
$$\int_A |f^{-1}(y)| dy = \int_B \det df dx \leq \liminf_{n\to\infty} \int_B …
- 2,504