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Results tagged with ap.analysis-of-pdes
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user 88462
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
2
votes
0
answers
300
views
About definition of NTA domain
I'm not an expert in analysis on very rough domains, such as NTA(Nontangentially Accessible Domain).
Here is my question. Usually, NTA domain $\Omega$ is a domain that has inner and outer corkscrew co …
- 323
2
votes
0
answers
106
views
Reference request : Global boundedness of weak solution for Neumann problem
I have some question on global boundedness of weak solution to Neumann problems.
Let $u\in W^{1,2}(\Omega)$ is a weak solution for Neumannn problem
$$ \mathrm{div} (A \nabla u )= \mathrm{div}\, F\quad …
- 323
2
votes
1
answer
262
views
Weak-star approximation of smooth functions in weak $L^p$-space
It is well known that the weak space $L^{p,\infty}$ has less density property contrary to standard $L^p$ space. Related to this one, I'm struggling to prove the following statement which is given in t …
- 323
3
votes
0
answers
79
views
Reference requests: $W_{p}^1$-estimate for $(\triangle -\lambda)$ on Lipschitz domains
Let $1<p<\infty$ and $\lambda>0$. When $\Omega$ is a bounded $C^1$ or a bounded Lipschitz domain
with small Lipschitz constant in $\mathbb{R}^d$, then for every $f\in L_p(\Omega)$ and $\mathbf{F}\in L …
- 323
1
vote
Does $\int_0^t \Vert u_x(s,\cdot) \Vert_{L^2} ds \le C$ imply $\Vert u_x (t,\cdot) \Vert_{L^...
In the case of linear heat equation, we write
$$ u(x,t) = \int_{\mathbb{R}} \Gamma(x-y,t) u_0(y) dy.$$
Here $\Gamma(x,t)$ is the standard heat kernel. Note that
$$ \left|\left(\frac{\partial}{\partial …
- 323
2
votes
0
answers
87
views
Solvability of Neumann boundary problems with singular boundary data $g \in (H^{1})^{*}$
I have a question on the solvability of Neumann boundary problems with singular data. To state my question, let $\Omega$ be a bounded Lipschitz domain (open and connected) in $\mathbb{R}^n$.
In the …
- 323
5
votes
1
answer
278
views
de Rham theorem for tempered distributions
I am wondering if the following statement holds.
If $u\in \mathscr{S}'$ satisfies $\left< u,\Phi\right>=0$ for all $\Phi \in \mathscr{S}$ with $\mathrm{div}\, \Phi=0$, then there exists $p\in \mathsc …
- 323
3
votes
1
answer
447
views
An inequality from Bessel potential space to Besov space
I'm not sure this question is suitable for MathOverflow. Currently, I'm reading a paper "Inhomogeneous Dirichlet Problem in Lipschitz domain" by Jerison and Kenig.
I have a question on some inequalit …
- 323