All Questions
1,304 questions
2
votes
1
answer
352
views
Poisson equation on noncompact manifold
Let $(M,g)$ be a complete non-compact manifold with bounded geometry, such that the Sobolev embeddings hold and $C^\infty_c$-functions are dense in $L^p_k$ space.
For the equation
$$\Delta u=f,$$
...
- 1,915
2
votes
0
answers
169
views
A basic question about the Spectral Theorem
Let $\Omega$ be a bounded open region in $\mathbb{R}^n$ and $\phi_i $ be the eigenfunctions of $-\Delta$ with Dirichlet boundary condition, i.e.
$$-\Delta \phi_i=\lambda_i \phi_i, \ \ \phi_i|_{\...
- 434
7
votes
1
answer
311
views
Almost orthonormal projection and orthonormal projection in Hilbert space
Let $(e_i)_i$ be a family of vectors in a Hilbert space being almost orthonormal but not quite, i.e.
$$\langle e_i, e_j \rangle \approx \delta_{i,j} + \alpha e^{-\vert i-j \vert} $$
and $\alpha$ is ...
- 71
1
vote
1
answer
296
views
Boundary behavior of Greens functions on smooth bounded (planar) domains
It is well known that for any smooth bounded (connected) domain $\Omega\subset\mathbb R^d$ with $d\ge2$, we can define a Green's function $G:\Omega\times\Omega\to\mathbb R$ in $\Omega$ which is smooth ...
- 1,247
1
vote
0
answers
40
views
Various definitions of weak solutions and continuity property
For example, consider the transport equation
$$
\partial_tu+b\cdot\partial_xu=0, u(0,x)=u_0(x),
$$
where $t\ge0,x\in\mathbb{R}^3$, $b\in W^{1,1}_{\rm{loc}}([0,\infty)\times\mathbb{R}^3)$.
It seems ...
- 71
7
votes
1
answer
337
views
Flows in Hilbert spaces
Let $\varphi: [0,T] \rightarrow H$ be a Hilbert space valued $C^1$-function. Let $H = X \oplus X^{\perp}$ such that $\varphi(0) \in X$ and the implication $\varphi(t) \in X \Rightarrow \varphi'(t) \in ...
- 83
0
votes
1
answer
233
views
Existence of subsequences convergence with weak topology
Let $\left\{ {{\varphi _n}} \right\}$ is the sequence bounded in ${L^\infty }\left( {0,\infty ;H_0^1\left( {0,1} \right)} \right)$. Is there exists $\varphi \in {L^\infty }\left( {0,\infty ;H_0^1\...
- 183
3
votes
1
answer
212
views
Eigenvalue estimates for operator perturbations
I edited the question to a general mathematical question, since I found the answer in Carlo Beenakker's reference and think that my initial question was mathematically misleading.
What was behind ...
- 536
4
votes
0
answers
747
views
Maximum Principles in Parabolic PDE with Neumann Condition
I am looking for some maximum principles and comparison results for parabolic equations. The most complete book I've found on this subject is: Murray Protter, Hans Weinberger - Maximum Principles in ...
- 1,759
4
votes
1
answer
597
views
Meaning of Alberti rank-one theorem
Heuristically what does Alberti's rank-one theorem imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$?
Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ ...
user123457
4
votes
0
answers
164
views
What's the essential definition of resonance of Schrodinger operator?
Rencently, I am reading some articles about time decay estimates or Strichartz estimates for Schrodinger equations with potential. When considering Strichartz estimates for potential $V$ with decay $|...
- 429
4
votes
1
answer
129
views
Meaningful generalization of viscosity solutions to higher order equations
Is there a meaningful generalization of the notion of viscosity solutions to third and fourth order equations?
user75426
2
votes
0
answers
61
views
Uniqueness of solution to Cauchy problem with quadratic nonlinearity
Consider the non-linear differential operator
$$\mathfrak{L}: \ C^2((0,T)\times\mathbb{R}^2)\ni\varphi\equiv\varphi(t,x,y) \, \mapsto \, \partial_x^2\varphi + (\partial_x\varphi)^2.$$
For $U\subset\...
- 51
2
votes
1
answer
475
views
Question on definition of Dirichlet to Neumann operator
Assume $\Omega$ is an open, bounded subset of $\mathbb R^3$ with a
$C^2-$ boundary $\partial \Omega= \Gamma$. For $f \in
H^{1/2}(\Gamma)$, let $F \in H^1(\Omega)$ denote the weak solution of
the ...
- 227
2
votes
1
answer
563
views
Density in fractional Sobolev space
Suppose $s∈(0,1)$, $D$ is an open set in $\mathbb{R}^d$. Define
$$
H^s=(1−\Delta)^{-s/2}L^2\left(\mathbb{R}^d\right),
$$
$$
H^s_D=\left\{f\in H^s:f=0 \mbox{ a.e. on } D^c\right\}.
$$
Q: Is $C^\...
- 515
2
votes
1
answer
391
views
Entropy solution for linear transport equation
Consider the transport equations
$$ (1) \qquad \partial_t u + \operatorname{div}(bu) = 0$$
and
$$ (2) \qquad \partial_t u + b \cdot \nabla u= 0$$
Can we define a notion of entropy solutions for (1) ...
- 839
1
vote
0
answers
166
views
Solution to Heat Equation By Projection
Let $X\subseteq C(\mathbb{R}\times [0,\infty))$ be the collection of all solutions to the IV heat equation
$$
\partial_t u(t,x) = \partial^2 u(t,x) \qquad u(0,x)=p(0,x),
$$
for some fixed $p\in C^2(\...
- 5,405
1
vote
0
answers
136
views
Topology on function spaces for pointwise convergence
Suppose I have some collection of maps $T_\lambda: C^\infty(\Omega)\rightarrow C^\infty(\Omega)$ which are linear and parameterized by a parameter $\lambda >0$. (Perhaps more generally take $C^\...
- 11
2
votes
1
answer
258
views
$L^2$ bound and Sobolev spaces
Let $f \in L^2(\mathbb R)$ be a function such that
$$\vert f \vert_{\alpha}:=\sup_{h>0}h^{-\alpha}\Vert f(\bullet+h)-f \Vert_{L^2}< \infty$$
for some $\alpha \in (0,1).$
I would like to know ...
user69109
3
votes
0
answers
82
views
Compatibility between the source and the boundary condition for an Helmholtz-type equation
Let $\Omega$ an open, convex, bounded domain in $\mathbb{R}^3$, and let us fix also $z\in\mathbb{C}\setminus\mathbb{R}$. Given $\phi\in H^{3/2}(\partial\Omega)$, I would like to show the existence of ...
- 943
1
vote
0
answers
42
views
Energy estimate for $\theta_t + H(\theta)_x = 0$ in $t>0, x >0$?
Consider the IBVP for $$\theta_t + H(\theta)_x = 0, \qquad t>0, \ x>0$$ with $$H(\theta) = \frac{1}{\pi} \text{pv}\int_{0}^\infty \frac{\theta(y)}{y-x} dy$$
with Dirichlet boundary conditions. ...
- 161
3
votes
1
answer
1k
views
Gagliardo-Nirenberg inequality for bounded domain
For concreteness let's assume that $u\in W^{1,2}(\Bbb R^2).$ It is well known that
$$
\|u\|_4\le C \|u\|_2^{\frac 12} \|\nabla u\|_2^{\frac 12}.
$$
This is also true if $u\in W^{1,2}_0(\Omega)$ for a ...
- 1,245
2
votes
0
answers
89
views
Wave equation for smooth Schwartz kernels
Let $(M,g)$ be a closed Riemannian manifold, $D$ an essentially self-adjoint differential operator on $L^2(M)$. Then the operator group $\{e^{itD}\}_{t\in\mathbb{R}}$ formed via functional calculus ...
- 1,913
4
votes
1
answer
166
views
Compute $ \partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx$ in the sense of distributions where $u$ solves a PDE
Let $u:\Omega\subset \mathbb R^N \to \mathbb R$ be bounded function that solves an evolution PDE $\partial_t u(t,x)= L(u(t,\cdot))(x)$, where $L$ is some elliptic operator.
How can I compute the ...
- 109
5
votes
2
answers
470
views
Improving equi-integrability for a family $\mathcal F$ in $L^1(\Omega)$
Let $\mathcal F$ be a weakly compact subset of $L^1(\Omega)$. Dunford–Pettis theorem says that $\mathcal F$ is uniformly integrable. Also, by de la Vallée-Poussin theorem we can find an increasing ...
- 1,245
10
votes
1
answer
586
views
Nonlinear Schrödinger equation with discrete Laplacian
In the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$" by Colliander, Keel, Staffilani, Takaoka and Tao it is argued in the beginning ...
user127411
4
votes
0
answers
170
views
Pointwise convergence of the eigenfunctions expansion of $f(x)=\frac{1}{|x|}$
Let $\Omega\subset \mathbb{R}^n$ a bounded domain with smooth boundary, $0<\lambda_1\leq \lambda_2 \leq \dots \leq \lambda_k\leq \dots$ the Dirichlet eigenvalues and $\{w_k\}_{k=1}^{+\infty}$ an $L^...
user39481
1
vote
1
answer
430
views
Generator of an analytic semigroup of operators
I have an operator of the following form:
$$ A = \begin{bmatrix} 0 & h_1 & h_2 \\ 0 & \Delta& h_3 \\ 0 & 0 & h_4 \end{bmatrix} $$
which results from a coupled PDE-ODE system ...
- 11
1
vote
0
answers
922
views
A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it
Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
- 698
3
votes
1
answer
876
views
Is Quantum Mechanics (norm)-consistent?
I edited a few small comments to the question in order to make it perhaps more comprehensible.
Today I came across the following question in quantum mechanics.
In Quantum mechanics it is common to ...
- 39
5
votes
0
answers
438
views
Green's formula and traces in weighted Sobolev spaces
Let $B_1$ denote the unit ball in $\mathbb{R}^d$, let
\begin{equation}
\rho(x) = 1-|x| \quad \text{ for } x \in B_1,
\end{equation}
and let $\sigma >0$ be given. As per the comments, notice that $\...
- 309
2
votes
1
answer
150
views
Two PDE for one unknown?
Let $x \in (0,L)$, $t \in (0,T)$, and let $f_1 = f_1(x,t) \in \mathbb{R}$, $f_2 = f_2(x,t) \in \mathbb{R}$, $u^0 = u^0(x) \in \mathbb{R}$ and $g= g(t) \in \mathbb{R}$ be continuous functions.
My ...
- 115
1
vote
1
answer
250
views
Moser/Schauder estimates for coercive boundary conditions
Consider the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on $(0, \infty) \times \Omega$, where $\Omega \subset \mathbb{R}^n$ is an open bounded domain with smooth boundary, and $L$ is a ...
- 11
1
vote
1
answer
219
views
Harmonic functions vanishing on the boundary and distance function asymptotics
Let $\Omega \subset \mathbb R^N$ be a $C^2$ domain. Let $u$ be a function such that $u \in W^{2,2}(\Omega)$ and $u = \Delta u = 0$ on $\partial \Omega$. Is it true that $$ c \le \frac{u}{[\mathrm{dist}...
user139845
1
vote
0
answers
102
views
Commutator estimates for $-(-\Delta)^s$, with $s \in (1,2)$
I'm currently trying to work with the non-local operator given by
$$
(-\Delta)^{\frac{s}{2}}f(x)= c_s\text{P.V} \int_{-\infty}^\infty \frac{-f(x+y)-f(x-y)+2f(x)}{|y|^{1+s}} dy,
$$
where $f :\mathbb ...
- 446
5
votes
1
answer
3k
views
How to learn concepts of Functional Analysis which are common in PDE
I am a master student and working in PDE area. I am trying to gain deep understanding of some of the concepts in functional analysis which are common tools in PDE research, such as weak*-topology, ...
- 371
0
votes
1
answer
419
views
Stone–von Neumann theorem?
The Stone–von Neumann theorem says that given two unitary groups on a Hilbert space $H$ satisfying the canonical commutation relations (CCR)
$$
U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t
$$
...
3
votes
0
answers
376
views
Existence and uniqueness for reaction-diffusion equations
I am interested in the following PDE on a $d$-dimensional torus $\mathbb{T}^d$
\begin{align*}
&\partial_tu(t,x) = \Delta u(t,x) +f(u(t,x),t,x),\\
& u(0)=u_0\in L_2
\end{align*}
where the ...
- 931
2
votes
1
answer
102
views
Equivalence of different topologies of Kernel of constant coefficient hypoelliptic operator
Let $P(D)$ be hypoelliptic operator with constant coefficients in $\mathbb{R^n}$.Let $\Omega$ be an open subset of $\mathbb{R^n}$ and $\mathscr{N_\Omega}$ denote space of distribution solutions of ...
3
votes
1
answer
283
views
Question on relation between a parabolic sobolev space and a sobolev bochner space
For parabolic sobolev spaces I follow the following definition:
According to this definition, we have that $W^{1,1,2}(I \times \Omega)=L^2(I; W^{1,2}(\Omega)) \cap W^{1,2}(I; W^{-1,2}(\Omega))$
...
- 227
2
votes
0
answers
181
views
Some questions about convergence
I'm getting lost in the first couple of sentences of a paper by Cafarelli & Alt. The sections are organized as follows, 1.1 Data, 1.2 The problem, 1.3 the statement and relevant part of proof.
1.1 ...
- 427
1
vote
1
answer
488
views
Motivation behind the parabolic metric
I've been reading some papers about parabolic evolution problems between manifolds. We want to study the behaviour of maps from the domain $(0,T)\times \Omega$, where $\Omega\subset \Bbb R^m$, to some ...
- 1,245
2
votes
0
answers
159
views
Principal symbol of a non-local operator and Atiyah–Singer index formula
I am trying to understand the Atiyah–Singer index formula for pseudo-differential operators. As far as I understood, the Fredholm index of the operator $A$ on a manifold can be computed just from the ...
- 41
7
votes
0
answers
433
views
(geodesic) smoothness of f-divergence with respect to the Wasserstein metric
We consider the f-divergence, which takes the form
$$
D_f(P \| Q) = \int_\Omega f\left(\frac{dP}{dQ}\right) dQ.
$$
For example, when $f(t) = t \log t$, we obtain the KL-divergence.
My question is ...
- 1,127
1
vote
0
answers
64
views
Bound on number of linearly independent eigenvectors of adjoint of composition operator
Fix $N>1$. Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that the composition operator via
$$
\begin{aligned}
C_f:C^{\infty}(\mathbb{R},\mathbb{R}) &\rightarrow C^{\infty}(\mathbb{R},\...
- 5,405
2
votes
1
answer
1k
views
Proof of Agmon's inequality in $\mathbb{R}^3$
According to Wikipedia, Agmon's inequality provides a bound on the $L^\infty$ norm of a $H^2$ function on a (regular) subset of $\mathbb{R}^3$. In the book of JC Robinson et al. "The Three-...
- 93
8
votes
3
answers
884
views
abstract evolution equations
Hi
Whenever I read a book on evolution equations, they set up, say the parabolic PDE
$$\dot{y} = Ay + f$$
in abstract function spaces (eg. $L^2(0,T;V)$ and $L^2(0,T;V^*)$). In examples, they always ...
- 107
2
votes
1
answer
382
views
Is the translation/dilation of an $L^p$-multiplier again an $L^{p}$-multiplier?
Suppose that $m:\mathbb R \to \mathbb C$ satisfies: there exists $C > 0$ such that
$$\| (m \hat{f})^{\vee} \|_{L^{p}} \leq C \|f\|_{L^{p}}.$$
That is, $m$ is an $L^{p}$-multiplier. Let $M(L^{p}...
- 1,051
5
votes
1
answer
472
views
Embedding theorem for anisotropic Sobolev spaces
Let $d=d_1+d_2$, $s_1,s_2>0$, $p>1$ and $(x_1,x_2)\in \mathbb{R}^{d_1}\times \mathbb{R}^{d_2}$, $(\xi_1,\xi_2)\in \mathbb{R}^{d_1}\times \mathbb{R}^{d_2}$. Define
$$
W^{s_1,s_2}_{p}:=\left\{f: ...
- 515
4
votes
1
answer
398
views
Proving the inequality $|\nabla |\nabla^r \psi|| \le |\nabla^{r+1} \psi|$
Following Aubin's book "Some nonlinear problems in Riemannian geometry", we use the notation
$$
|\nabla^r \psi|^2 = \nabla_{\alpha_1}\cdots \nabla_{\alpha_r}\psi \nabla^{\alpha_1}\cdots \nabla^{\...
- 1,245