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Results tagged with ap.analysis-of-pdes
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user 42047
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
1
vote
Accepted
Strong Maximum Principle for very weak supersolutions of Laplacian operator
As such the statement does not make sense: if $u \in L^1_\mathrm{loc} (\Omega)$, its restriction on $\partial \Omega$ is not well-defined.
If $u \ge 0$ on $\Omega \setminus K$, where $K \subset \Omeg …
- 2,824
2
votes
Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$
I your aim is to apply the Galerkin method, you do not need simultaneous orthonormal basis.
An inspection of Evans’ proof shows that you need a sequence of linear maps $(P_n)_{n \in \mathbb{N}}$ such …
- 2,824
3
votes
Accepted
Why $\|u\|_{\tau}\leq C[u]_{W^{s,p}}^a\|u\|_{L^q}^{1-a}$ not correct for $p=1$?
The issue with $p = 1$ is that some definitions of fractional Sobolev spaces that were equivalent when $p > 1$ (by the Gagliardo seminorm that you gave, by interpolation between functional spaces, by …
- 2,824
2
votes
Friedrichs/Poincare inequality on $S_n \times (0,\infty)$?
By Fubini's theorem and by taking functions that do not depend on the $\mathbb{S}^n$ variable, your estimate is equivalent with the inequality
$$
\int_0^\infty \vert u \vert^2 \le C \int_0^\infty \v …
- 2,824
3
votes
"Schwarz symmetrization" on annulus
Such a construction is not possible.
One way to see this is to note that if such a construction was possible, then you would have, by the embeddings for Sobolev spaces of radial functions
$$
\Vert …
- 2,824
1
vote
Accepted
$u_n$ bounded in $L^\infty(0,T;H) \cap L^2(0,T;V)$ implies $u_n \to u$ strongly in $L^2(0,T;...
As it is stated, this property does not hold: indeed consider the sequence of functinos $(u_n)_{n \in \mathbb{N}}$ defined for $t \in [0, T]$ by
$$
u_n (t) = \sin (2n\pi t) v,
$$
where $v \in V$ is …
- 2,824
2
votes
Accepted
Sobolev's lemma on manifolds
This follows from its counterpart in the Euclidean space by local charts.
If you want to have an estimate on the derivative $D^r f$, then you should impose some bounds on derivatives of the curvatur …
- 2,824
2
votes
Sobolev trace map: is the fractional seminorm bounded by just the gradient?
This should follow from the nonhomogeneous trace inequalty
$$
\vert T u \vert_{H^{1/2} (\partial M)} \le C \lVert \nabla u \rVert_{L^2} + \lVert u \rVert_{L^2},
$$
and and from the classical Poincar …
- 2,824
3
votes
traces of sobolev spaces under additional assumptions
Partial answer: according to Triebel (Theory of function spaces, 1983, Remark 2.7.5, p. 139), the trace of the Besov space $B^{1/p, p}_1 (\Omega)$ is $L^p (\partial \Omega)$, but the linear extension …
- 2,824
10
votes
Differential of a Sobolev map between manifolds
If you are interested, we have given an intrinsic definition of a weak derivative for maps between manifolds A. Convent et J. Van Schaftingen, emphasized Intrinsic colocal weak derivatives and Sobolev …
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4
votes
Equivalent Norms on Sobolev Spaces
If $k \in (0, 2]$, we define the multiplier
$$
m (\xi) = (1 + \vert \xi \vert^2)^\frac{k}{2} - \vert \xi \vert^k.
$$
We observe that if $\vert \xi \vert \ge 2$, then by differentiability
$$
\big …
- 2,824
1
vote
Bounded input Bounded output stability for heat equation
In the paper by Weidermann, Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$−norm, 2002, you can find in particular the estimate
$$
…
- 2,824
2
votes
Accepted
Hardy-type inequality for point boundary
If $p > n/2$ and if $f \in C^2_c (\mathbb{R}^n \setminus \{0\})$ (twice continuously differentiable functions whose support is compact in $\mathbb{R}^n \setminus \{0\}$), then the weighted Hardy inequ …
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5
votes
Accepted
Reference request: Riesz potential $I_\alpha : L^{d/\alpha} \to \rm{BMO}$?
Let $a \in \mathbb{R}^d$ and $r > 0$. We have
$$
\frac{1}{\vert B_r \vert^2}
\int_{B_r} \int_{B_r} \vert I_\alpha (f) (x) - I_\alpha (f) (y) \vert\,\mathrm{d}x\,\mathrm{d}y
\le \frac{c_{d, \alpha} …
- 2,824
4
votes
Example for the Sobolev embedding theorem when p=n.
You can take as an example
$$
u(x) = x_1^{k - 1} (\log \lvert x \rvert)^\beta:
$$
if $\beta < 1 - \frac{1}{n}$, $u \in W^{k,n} (B_1)$ and if $\beta > 0$ then $D^{k - 1} u \not \in L^\infty (B_1)$.
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