All Questions
1,304 questions
3
votes
1
answer
531
views
An argument in the proof of a compactness theorem
In the proof of a compactness theorem involving fractional derivatives in Temam's Navier-Stokes Equations, an argument as the following is made.
Suppose $X_0,X,X_1$ are Hilbert spaces such that
...
CommunityBot
- 1
1
vote
0
answers
105
views
compactness of sequence of harmonic functions
Let $ \Omega$ denote a smooth bounded domain in $ R^N$ and let $u_m \in C^\infty( \overline{\Omega})$ harmonic functions. We also suppose $ u_m$ is bounded in $L^2(\Omega)$ (uniformly in $m$).
...
- 1,385
1
vote
0
answers
331
views
Verifying a claim regarding $H^1$ weak convergence and $L^2$ strong convergence on a surface
I'm reading a paper whose first section discussed $H^1$ maps defined on star-shaped sets, but I got stuck in verifying a claim for quite a while. I'm now thinking the claim is wrong, but it's hard to ...
- 1,350
2
votes
0
answers
64
views
The continuity of $L^2$ gradient on moving domain
I post this on MSE too. Sorry for this short cross posting. But I stuck in this problem for a while... I don't know what the right tool should be used to handle the moving domain problem...
Let $I:=(...
CommunityBot
- 1
1
vote
0
answers
128
views
determine when $e^{ikx}$ can be boundary value of a holomorphic function
Assume that $\Gamma=\{x+if(x): x\in \mathbb{R}\}$ is a graph, separating $\mathbb{C}$ into two connected components. Let's denote the one below $\Gamma$ by $\Omega$.
My question is, for what curves $...
- 24.2k
8
votes
2
answers
634
views
Existence of a uniformly continuous function $g$ on $\mathbb{R}$ where $f = g$ a.e.?
Suppose $f \in L^\infty(\mathbb{R})$, $f_h(x) = f(x + h)$, and$$\lim_{h \to 0} \|f_h - f\|_\infty = 0.$$Does there exist a uniformly continuous function $g$ on $\mathbb{R}$ such that $f = g$ almost ...
- 3,709
2
votes
1
answer
755
views
Existence of a solution to an infinite dimensional Stratonovich SDE
Let
$U,H$ be separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with finite trace
$U_0:=Q^{1/2}U$
$(\Omega,\mathcal A,(\mathcal F_t)_{t\ge 0},\operatorname P)$ ...
CommunityBot
- 1
2
votes
0
answers
77
views
When do finite dimensional approximations approximate the spectral absicssa of a linear operator?
I apologize if the following is trivial for experts in the field. If so, please feel free to refer me instead to any proper references.
I would like to compute the spectrum of a known non-normal, ...
- 121
5
votes
1
answer
227
views
On the 'usefulness' of the 'original' definition of viscosity solution
In [CEL84, Theorem 1.1, p.489], Crandall, Evans, and Lions give three equivalent definitions of viscosity solution.
As the authors note, the first two are "more appealing in some respects and more ...
- 331
2
votes
1
answer
367
views
Equivalence of alternative definitions of 'viscosity solution'
Consider the first-order Hamilton-Jacobi equation (HJ):
$$H(x,u,\nabla u) = 0 \quad \text{ on } \ \Omega,$$ where $\Omega$ is an open set of $\mathbb{R}^n$, $u:\Omega \to \mathbb{R}$, and $H:\Omega \...
- 331
2
votes
0
answers
178
views
are these norms equivalent?
If it is known that $\sum_{i,j=1}^{n}a_{ij}\xi_i\xi_j\geq \alpha^2|\xi|^2$, where $\xi=(\xi_1,\xi_2,...,\xi_n)\in\mathbb{R}^n$ then can it be said that $\sum_{i,j=1}^{n}a_{ij}\frac{\partial u}{\...
- 157
2
votes
1
answer
158
views
Defect of Compactness for the Strichartz Estimates
I am trying to understand the motivation behind the main theorem of Keraani: See the top of page 356 and which is the motivation behind his Theorem 1.6
We consider
$$i\partial_t u + \Delta u =0, u(x,...
- 39.1k
0
votes
0
answers
343
views
A question on weak formulation of the p-laplacian operator
Can it be said that $$\int_{\Omega}\Delta_p u |\phi|^{p-2}\phi dx=\int_{\Omega}\Delta_p \phi |u|^{p-2}u dx\qquad\forall \phi\in C_0^2(\overline{\Omega})$$ is the generalized weak formulation of $$\...
- 4,195
4
votes
0
answers
2k
views
Fourier transform of $C^\infty_0$, smooth functions vanishing at infinity
Is there a proper description of the space $$\{\hat f\ | \ f\in C^\infty \ s.t. \forall \alpha\in\mathbb{N}^n,\forall \epsilon>0\exists K\subseteq \mathbb{R}^n\ K\ \text{compact};\ \sup_{x\in \...
- 41
0
votes
1
answer
445
views
Domain of the Stokes operator
Let
$\Omega\subseteq\mathbb R^d$ be open ($d\in\mathbb N$)
$\mathcal D:=C_c^\infty(\Omega)^d$ and $$\mathfrak D:=\left\{\phi\in\mathcal D:\nabla\cdot\phi=0\right\}$$
$\mathcal H:=\overline{\mathfrak ...
CommunityBot
- 1
5
votes
1
answer
4k
views
Difference between the dual space of $H^1(\Omega)$ and the dual of $H^1_0(\Omega)$
This is cross-posted on MSE: https://math.stackexchange.com/q/1596565/9464
In the Partial Differential Equations by Evans (2nd edition p299), $H^{-1}(\Omega)$ denotes the dual space to $H^1_0(\Omega)$...
CommunityBot
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3
votes
2
answers
397
views
The space $L^p(\partial\Omega)$ in cited references
The space $L^p(\partial\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$ appears in a lot of PDE textbooks without being given any definitions, not even in those with a detailed appendix ...
- 215
1
vote
0
answers
90
views
Monotone operator subgradient
I am trying to solve a PDE of the form $\mathcal{A}u'(t) + \partial\Psi[u(t)] \ni 0$ where $\mathcal{A}$ is a skew-symmetric, linear, monotone operator, $\Psi$ is convex, and $\partial \Psi$ is the ...
- 11
3
votes
1
answer
187
views
Free quantum evolution operator on Sobolev space
I am not a mathematician, but would like really like to get some confirmation on the things I am doing here.
Let $-\Delta: H^2(\mathbb{R}) \subset L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ then ...
- 95
2
votes
3
answers
979
views
"Must read" papers in functional analysis and PDE (à la Trefethen) [closed]
The following question has been inspired by "Must read" papers in numerical analysis.
In 1993, Prof. Trefethen published a NA-net posting with a list of thirteen papers that he had used in ...
CommunityBot
- 1
2
votes
1
answer
703
views
Correction term in the relation between the Itō and Stratonovich integrals in Hilbert spaces
I'm reading the paper On the relation between the Itō and Stratonovich integrals in Hilbert spaces and there is something I don't understand.
In the notation of the paper, let
$H,H_1$ be separable $\...
CommunityBot
- 1
3
votes
1
answer
351
views
Gradient zero a.e on the the zero set
In Brezis Functional Analysis Page 314 Point 4 it is given that for u in $W^{1,p}(\Omega)$ where $\Omega$ is any open set then $\nabla u=0$ a.e on the set where $\{u(x)=k\}$, k is a constant.
How ...
- 29.8k
0
votes
1
answer
247
views
Gradient bounds on Newtonian potentials
Suppose $N \ge 3$ and let $\Phi(x):= C_N |x|^{2-N}$ is the fundamental solution. Let $\Omega$ denote a bounded domain in $ R^N$.
Consider $ -\Delta u(x) = f(x) $ in $\Omega$ with $u=0$ on $ \...
- 201
1
vote
1
answer
1k
views
Introductory text to Sobolev spaces and PDE's [closed]
I'm looking for a good introductory to Sobolev, preferably with an emphasis to their relationship to PDE's analysis.
I have only seen thus far Giovanni Leoni's "First Course in Sobolev Spaces" which ...
- 579
2
votes
0
answers
116
views
The composition of a dissipative operator and a positive operator is dissipative?
Consider the following bilinear system on a open and bounded domain $\Omega$
\begin{equation}
\left\{\begin{array}{r c l}
\displaystyle\frac{dy(t)}{dt} &=& Ay(t)+u(t)By(t)\\
y(0) &...
- 630
2
votes
1
answer
219
views
Right inverse of the Seiberg-Witten functional
For closed 4 manifold X, we consider the derivative of the Seiberg-Witten functional, i.e.
$$\Omega^1_2(X;\sqrt{-1}\mathbb R)\oplus\Gamma_2(S^+)\overset{D}{\to}\Omega^2_{+,1}(X;\sqrt{-1}\mathbb R)\...
- 17.5k
0
votes
0
answers
119
views
Gauge Fixing Problem on Cylindrical
For Cylindrical $Y\times\mathbb R$, where $Y$ is a closed oriented 3-manifold.
If it is necessary, we could consider the $b_1(Y)=0$ case.
Fix a Line bundle $L\to Y\times \mathbb R$ and a Hermitian ...
- 1,915
1
vote
0
answers
117
views
The eigenfunction of modified $1$-laplace equation?
Let $\Omega\subset \mathbb R^2$ be open bounded with smooth boundary. It is well known that the laplace equation
$$
-\Delta u=0
$$
has a set of eigenvalues $0<\lambda_1<\lambda_2\leq\lambda_3<...
- 679
2
votes
2
answers
953
views
Differentiability of Nemytskii operator on Sobolev space
I am trying to consider hypothesis on $g$ such that the operator
$$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$
is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...
- 3,584
2
votes
0
answers
110
views
If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$
Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$.
Given then $f\in L^1(\Bbb R^n)$; by density there exists $\{f_j\}_j\...
CommunityBot
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2
votes
0
answers
86
views
I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection
I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
CommunityBot
- 1
4
votes
1
answer
364
views
$H^1$-continuity of Laplace's equation with respect to boundary data
Let $\Omega\subset \mathbb{R}^d$ be open and bounded with $C^\infty$ boundary $\partial\Omega$, $\phi\colon \partial\Omega \rightarrow \mathbb{R}$ continuous and $u^\phi$ the solution to Laplace's ...
- 13k
6
votes
1
answer
322
views
finding subharmonic function on the ball with both Dirichlet and Neumann boundaries prescribed
I have a question which looks like some sort of inverse problem.
Let $B$ denote the unit ball centered at the origin in $R^N$ (take $N \ge 2$).
Given any $h:\partial B \rightarrow (0,\infty)$ (smooth) ...
- 4,899
8
votes
1
answer
502
views
Are smooth solutions to a PDE dense in the space of $L^2$ solutions to the PDE?
Let's say I have a linear differential operator $P$ with smooth coefficients between bundles $E$ and $F$ over a smooth compact manifold $X$ with smooth boundary. Let's consider $P$ as an operator ...
CommunityBot
- 1
5
votes
1
answer
991
views
Boundary regularity of Dirichlet Eigenfunction on bounded domains
Consider a bounded, connected and open subset $\Omega\subset \mathbb{R}^d$ and the Dirichlet Laplacian $-\Delta$ acting in $L^2(\Omega)$.
Then we know that the eigenvalues of $-\Delta$ form an ...
CommunityBot
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7
votes
1
answer
344
views
Level sets of weakly differentiable funtions
Let $C$ be a $C^1$ hypersurface in $R^n$ and let $u \in C^1(R^n)$. Suppose
$$\nabla u(x) \cdot \eta(x)=|\nabla u| \ \ \forall x\in C$$
where $\eta(x)$ is the normal vector to $C$ at $x$ ($\nabla u$ ...
1
vote
1
answer
2k
views
Lipschitz functions and $W^{1,\infty}$
I am not sure my question is research type, but I am sure I can find here an answer.
So we have the following theorem in the book of Lawrence Evans in PDE, 2nd edition pages 294-295:
Theorem 4 (...
- 755
4
votes
1
answer
1k
views
Density argument with Schwartz functions?
I was wondering whether the Schwartz functions are also dense in
$$\{f \in L^2(\mathbb{R}^n); \int_{\mathbb{R}^n} |x|^2 |f(x)|^2 dx + \int_{\mathbb{R}^n}|\xi|^2 |\hat{f}(\xi)|^2 d \xi < \infty\}$$
...
- 24.2k
0
votes
0
answers
378
views
compact injection
Put:
$D=\{u\in L^{2}(\mathbb{R}^{n})| x^{\alpha}D^{\beta}_{x}u\in L^{2}(\mathbb{R}^{n}), \forall \alpha,\beta \in \mathbb{N}^{m}:|\alpha|+|\beta|\leq 2 \}$
Why $D \hookrightarrow L^{2}(\mathbb{R}^{n}...
- 39.1k
2
votes
1
answer
579
views
Is the Lopatinski-Shapiro condition invariant under diffeomorphism?
If a PDE (eg. the heat equation with Robin BCs, or the elliptic version) on a bounded smooth domain $U$ satisfies the Lopatinski-Shapiro condition (for a definition see eg. Wloka), and if $T:U \to W$ ...
- 5,343
3
votes
0
answers
95
views
Strengthening of the local smoothing estimates for the free Laplacian
The classical local-smoothing estimates for the free Laplacian asserts that:
$$\Vert e^{-it\Delta}f\Vert_{L^2((-\infty,+\infty)\,;\,H^{1/2}(B))}\leq C_B\cdot\Vert f\Vert_{L^2}$$
where $B\subset\mathbb{...
CommunityBot
- 1
7
votes
0
answers
304
views
Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories
Let
$d\in\left\{2,3\right\}$
$\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$
$\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...
CommunityBot
- 1
3
votes
0
answers
73
views
On the principal eigenvector of an elliptic operator
Suppose I have an open domain $U \subset \mathbb{R}^n$ and an elliptic operator $L$ acting on all square-integrable $C^2$ functions $\rho:U\to \mathbb{R}$ which converge to zero at $\partial U$:
\...
CommunityBot
- 1
4
votes
0
answers
172
views
Donnelly-Fefferman growth of eigenfunctions
Let $(M, g)$ be a compact Riemannian manifold, and let $\lambda^2$, $\varphi_\lambda$ represent eigenvalues and eigenfunctions respectively of the Laplacian $\Delta$, that is, $-\Delta \varphi_\lambda ...
- 41
2
votes
0
answers
282
views
Reference request: functional analysis results used in Taubes paper (1980)
I am studying Taubes paper 'Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations'. I am looking for a reference of three following theorems:
Let $f(x)$ be a convex funtional ...
- 6,259
1
vote
1
answer
378
views
Easy Garding Inequality
Easy Garding Inequality states that if $a=a(x,\xi)$ is a symbol in $S=\{a\in C^{\infty}||\partial_{\alpha}a|<C_{\alpha} \hspace{2mm} \forall \alpha\}$ with $a\geq \gamma >0 $ on $\mathbb{R}^{2n}$...
- 16.2k
1
vote
1
answer
401
views
The reproducing kernel for harmonics on compact manifolds
Page 39, proposition 1.1.3 here, http://www.cis.upenn.edu/~cis610/sharmonics.pdf clearly explains how for every ``level" (the parameter $k$ in the proposition) one can construct a function ("kernel") ...
- 23k
6
votes
1
answer
357
views
Travelling waves for nonlinear Schrödinger equation
Consider the following nonlinear Schrödinger equation:
$$
-\Delta \Phi - i\frac{\partial \Phi}{\partial t} = f(|\Phi|^2)\Phi,
$$
where $\Delta$ is the Laplacian on $\mathbb{R}^n$, $f$ gives the ...
CommunityBot
- 1
1
vote
0
answers
192
views
The decay rate of the spectrum of the Gaussian kernel on compact manifolds
It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...
- 617
5
votes
1
answer
3k
views
How to learn concepts of Functional Analysis which are common in PDE
I am a master student and working in PDE area. I am trying to gain deep understanding of some of the concepts in functional analysis which are common tools in PDE research, such as weak*-topology, ...
- 121