All Questions
Tagged with fa.functional-analysis ap.analysis-of-pdes
546 questions with no upvoted or accepted answers
4
votes
0
answers
199
views
Spectral problems with the wrong sign on the Poincaré disk
Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$ equipped with the Poincaré metric $g$ of negative scalar curvature $-1$. Denote by $\Delta_g = \mathrm{Tr}_g(\nabla^g d) = - d^{\ast_g} d$ ...
- 2,816
4
votes
0
answers
160
views
An estimate for the Benjamin-Ono equation from T. Tao's well-posedness paper
In https://arxiv.org/abs/math/0307289 (eq. (8)),
for a (smooth) solution of the equation $$u_t - uu_x + Hu_{xx} = 0$$
(where $H$ denotes the Hilbert transform) the following estimate is stated (...
- 839
4
votes
0
answers
174
views
Techniques for showing non-degeneracy results (PDE)
Motivation:
Consider the equation,
$$-\Delta u = u^p$$
in $\mathbb{R}^n$ for $n\geq 3$ and $p=2^*-1.$ Then we know that this equation has unique positive solutions given by functions of the form $U_{a,...
- 537
4
votes
0
answers
194
views
$L^\infty$ solutions for parabolic Neumann problem (heat equation)
Consider the heat equation on a (smooth) domain in $\mathbb{R}^n$ with homogeneous Neumann BCs:
$$u_t - \Delta u = f$$
$$\partial_\nu u = 0$$
$$u|_{t=0} = u_0$$
where $f \in L^p(0,T;L^r(\Omega))$ and $...
- 307
4
votes
0
answers
127
views
Algebra properties regarding Gevrey spaces: closed under multiplication
In page 24 of the paper Landau Damping: Paraproducts and Gevrey Regularity, the authors claimed an algebra property of Gevrey spaces, the formula (3.14), without giving a proof. So I'm asking for a ...
- 517
4
votes
0
answers
143
views
If theorem valid for compactly supported distribution then is it also valid for tempered distribution?
I have seen many theorem which Author wanted to prove for tempered distribution, but without saying anything proves for compactly supported distribution.
For instance,
Theorem: Any $A \in \Psi^{m}$ ...
- 309
4
votes
0
answers
146
views
Fourier transform without characters (Eigenfunctions of an operator)
Let's consider a very simple problem in quantum mechanics:
We have, in $\mathbb R,$ a potential barrier of the form
$$
V(x) = V_0 \mathbf 1_{[-a,a]}(x),
$$
where $\mathbf 1_{[-a,a]}$ denotes the ...
- 1,755
4
votes
0
answers
139
views
Smooth dependence of solution to elliptic pde depending on parameter
I have a question in mind but let me generalize it slightly.
Suppose I am looking at some pde like
$$-\Delta u + t u = f(u)$$ in $B_1$ (here $u=u(x)$) with $u=0$ on $ \partial B_1$ where $B_1$ is ...
- 1,385
4
votes
0
answers
220
views
improved regularization for $\lambda$-convex gradient flows
It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...
- 5,380
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
0
answers
126
views
Relationship between three different definitions of solutions for ODE with irregular coefficient
What is the difference between the notions of
Regular Lagrangian flow
Filippov solution
Caratheodory solution
of an ODE $\dot \Phi(t,x) = b(t,\Phi(t,x))$, with initial condition $\Phi(0,x) = x$, ...
- 839
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
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
0
answers
93
views
Conditions on the Hamiltonian of a classical system that yeild essentially self-adjoint quantum Hamiltonian
What are the conditions on the Hamiltonian of a classical system that under these conditions the quantum Hamiltonian obtained via Weyl quantization will be essentially self-adjoint in $L_2(\mathbb{R}^...
- 381
4
votes
0
answers
111
views
A reference for $\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u$
Let $\Omega$ be an open domain with nice boundary and $u\in W^{1,p}(\Omega)$. I believe that $|u|^p\in W^{1,1}$ with
$$
\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u
$$
but couldn't find a good ...
- 1,245
4
votes
0
answers
188
views
Sobolev embedding in an annulus
Is it possible to determine the best constant $S(\Omega, p)$ of the embedding $H^1_{0, r}(\Omega)$ to $L^{p+1}(\Omega)$ where $p>1$ and $\Omega=\{a<|x|<b\}.$
- 63
4
votes
0
answers
169
views
Inverse Laplacian and convolution in Albeverio's “Solvable Models in quantum mechanics”
I asked this question on math.stackexchange.com two weeks ago but got no answers so far and I got no clues from literature, so maybe someone here knows a reference. I hope it is ok to ask this ...
- 141
4
votes
0
answers
145
views
An embedding question: Morrey spaces
Question. If $u\in L^1$ and $Du$ is in the dual of the Holder space $C^\alpha$, then is it possible to say $u$ belongs to some Morrey space $L^{1, \delta}$?
- 43.2k
4
votes
0
answers
174
views
Constant in trace theorem for balls
Consider the standard open ball $B_r:=\left\{x ; \left\lvert x \right\rvert \le R \right\}.$
The trace theorem tells us any function in $W^{k,p}(B_r)$ can be restricted to a function $W^{k-1,p}(\...
user127450
4
votes
0
answers
254
views
Lower semi-continuity of integration
I've found many papers characterizing the weak lower semi-continuity of
$$
\Phi(u)\triangleq \int_{x \in \Omega} f(x,u(x))\,dx,
$$
on $L^2(\Omega)$ where $\Omega$ is a bounded subset of $\mathbb{R}^d$....
- 5,405
4
votes
0
answers
136
views
Davies' definition of elliptic operators in "Heat Kernels and Spectral Theory"
I am trying to find my way through Davies' book, and one of the difficult points is his choice of what "elliptic operator" means in his text. This is the first time that I encounter some of the ...
- 5,407
4
votes
0
answers
198
views
Relationships between fractional Sobolev space, Bessel spacse and Hajłasz–Sobolev space
It is known that for $\alpha\in(0,1)$ and $p>1$,
the fractional Sobolev space $W^{\alpha,p}(R^n)$ is defined by
$$
W^{\alpha,p}(R^n):=\{f\in L^p(R^n):\int_{R^n}\int_{R^n}\frac{|f(x)-f(y)|^p}{|x-y|^...
- 411
4
votes
0
answers
164
views
A modern reference for the "Intermediate Derivatives Theorem"
In the book Non-Homogeneous Boundary Value Problems and Applications I by Lions and Magenes, the Intermediate Derivative Theorem is stated as follows:
Intermediate Derivative Theorem: Let $X\subset ...
- 1,247
4
votes
0
answers
311
views
Some elementary decay estimates of solutions to the heat equation
Preliminaries: Let $u$ be the solution of the Cauchy problem for the heat equation with initial datum $u_0 \in L^1 \cap L^p$. Then I know that the following estimates hold:
$$\Vert u(t,\cdot)\Vert_{L^...
- 303
4
votes
0
answers
89
views
How can I can derive an explicit bound for the solution of the poisson's PDE?
i need some help on this question
Let $\Omega$ be an open subset of $\mathbb{R}^{2}$ (say a square) with
$\partial{\Omega} =\Gamma_{1} \cup \Gamma_{2} \cup\Gamma_{3} \cup\Gamma_{4}$. A structure ...
- 41
4
votes
0
answers
128
views
Positive and Negative parts of functions in Schrodinger $U_{\Delta}^{p}$ spaces
Let $H$ be a Hilbert space. For $1<p<\infty$, define the atomic space $U^{p}(\mathbb{R};H)$ as follows. We say that $a(t)$ is a $U^{p}$ atom if $\{t_{k}\}$ is a partition of $\mathbb{R}$, $a_{k}\...
- 873
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 $$\{\hat f\ | \ f\in C^\infty \ s.t. \forall \alpha\in\mathbb{N}^n,\forall \epsilon>0\exists K\subseteq \mathbb{R}^n\ K\ \text{compact};\ \sup_{x\in \...
- 41
4
votes
0
answers
172
views
Donnelly-Fefferman growth of eigenfunctions
Let $(M, g)$ be a compact Riemannian manifold, and let $\lambda^2$, $\varphi_\lambda$ represent eigenvalues and eigenfunctions respectively of the Laplacian $\Delta$, that is, $-\Delta \varphi_\lambda ...
- 41
4
votes
0
answers
500
views
Properties of the solution of the heat equation
Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention.
Note 2: This is my research problem, but the original problem ...
- 679
4
votes
0
answers
121
views
The best constant in Poincare-liked inequality in $BV$ and $BD$ space
This question has been posted on Math Stack exchange for a while and received no response. So I decide to move it here to get more attention.
Let $\Omega\subset \mathbb R^N$ be open, bounded and with ...
- 679
4
votes
0
answers
284
views
How can I calculate the adjoint of the wave operator $\square_{g}$ in $H^{k}$?
I have a three part question, which I could only received an answer for the first part here.
The Laplace-Beltrami operator is an operator which is the typical example of a self-adjoint operator in $L^...
- 101
4
votes
0
answers
551
views
$f,g , |f|f, |g|g \in A(\mathbb R) \ \text{(Banach algebra)} \implies \left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$?
Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra
$$A(\mathbb R):= \{f\in L^{1}(...
- 1,051
4
votes
0
answers
434
views
Scattering for rapidly decaying solutions of NLS
Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions
of the Nonlinear Schrödinger Equation" the following property.
Given the problem
\begin{equation}
\left\{
\begin{array}{rl}
...
- 403
3
votes
0
answers
196
views
Parabolic smoothing for semilinear PDE
Consider the semilinear energy-critical parabolic PDE in $\mathbb{R}^3$
\begin{align}
\partial_t u &= \Delta u + |u|^{4/(n-2)}u = \Delta u + u^5\\
u(0,x) &= u_0\in \smash{\dot{H}}^1(\mathbb{R}^...
- 537
3
votes
0
answers
108
views
A question on essentially self-adjoint differential operators of the type $\Delta=D^{\ast}D$
Let $(M,g)$ be a smooth (connected, complete, oriented) Riemannian manifold and let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a linear partial differential operator, which I view as an operator in $L^{2}(...
- 1,171
3
votes
0
answers
90
views
Sobolev embedding on a compact manifold without boundary
I am reading M. E. Taylor, "Partial Differential Equations III", Second Edition, Springer-Verlag, New York, (1996).
In chapter 13, section 2, in Prop. 2.3 and Prop. 2.4, one finds the ...
- 311
3
votes
0
answers
167
views
Bounding the $L^{p*}$ norm from below for functions satisfying a $p$-capacity estimate
If $1 \le p < n$, the $p$-capacity of a compact set $A \subset \mathbb{R}^n$ with respect to an open set $U$ containing it is defined as $$\text{Cap}_p(A, U) := \inf \left\{\int_U |\nabla u|^p \, ...
3
votes
0
answers
147
views
Embeddings of Bochner-Sobolev spaces with second time derivative
NOTE: I also asked this question here in MSE.
In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these ...
- 91
3
votes
0
answers
100
views
Rate of convergence of mollified distributions in Besov spaces with negative regularity
Given a standard mollifier $\rho_\delta$ and a distribution $ u \in B^\alpha_{ p, p}$ with $\alpha<0$, $p \in [1, \infty]$ and $B^\alpha_{p,p}$ is a not-homogeneous Besov space, I'm trying to prove ...
- 293
3
votes
0
answers
124
views
Estimating a solution to Euler-type ODE #2
This is a similar question to this but with a different ODE.
Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer, $R&...
- 969
3
votes
0
answers
128
views
Image of trace operator on $W^{2,1}(\mathbb{R}^2)$
It is known that for a domain $\Omega\subset \mathbb{R}^2$ with $C^1$ boundary $\partial\Omega$, that the trace operator is bounded and surjective from $T:W^{1,1}(\Omega)\to L^1(\partial\Omega)$.
For ...
- 989
3
votes
0
answers
94
views
Harmonic heat flow, formal and rigorous
Let $ (M,g) $ be a smooth Riemann manifold without boundary, $ S^{n-1} $ is an $ n $-dimensional sphere, and $ T>0 $. Consider a weak solution $ u:M\times[0,T]\to S^{n+1} $ of
$$
\partial_tu-\Delta ...
- 1,219
3
votes
0
answers
84
views
About the naturality of Krasnoselskii genus on Variational Methods
I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
- 131
3
votes
0
answers
153
views
Quasimode construction on a compact Riemannian manifold
Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several ...
- 131
3
votes
0
answers
41
views
Functional of fully nonlinear equations
Let $\left(\mathcal{M}, g_0\right)$ be a compact Riemannian manifold of dimension $n>2$ and denote by 'Ric' and $R$ respectively the Ricci tensor and the scalar curvature. The $k$-Yamabe problem is ...
- 471
3
votes
0
answers
111
views
What is known about the analytic continuation of Maz'ya's modified harmonic zeta function $\sum_{n=1}^{\infty} e^{-zH_n}$?
Question:
If we let $H_n = \sum_{k=1}^{n} \frac{1}{k}$ be the harmonic numbers then we can consider the modified zeta function
$$ f(z) = \sum_{n=1}^{\infty} e^{-zH_n } = \sum_{n=1}^{\infty} e^{-z(\ln(...
- 2,689
3
votes
0
answers
83
views
Embedding theorems for Dini continuous functions
Are there embedding theorems for the space of Dini continuous functions on a Euclidean domain, or even just on an interval? Ideally, I am looking for something like the classical Morrey inequalities ...
- 3,388
3
votes
0
answers
127
views
Number of spatial critical points of a solution to the heat equation in higher dimensions
I would like to know if the number of spatial critical points of a solution to the heat equation can increase. Given $u_0:\mathbb S^n\to\mathbb R$, let $u$ be the solution of the initial value problem:...
- 31
3
votes
0
answers
102
views
Can Sobolev space be characterized by spectral decomposition?
Consider a homogeneous Carnot group $\mathbb{G}$ with step $r$. Let $X_1,\cdots,X_m$ be the first layer of its Lie algebra. Denote by $\mathcal{L}=-\sum_{i=1}^m X_i^2$ the sub-Laplacian on $\mathbb{G}$...
- 561
3
votes
0
answers
111
views
Complex interpolation of Fourier-Lebesgue spaces and Lebesgue spaces
The complex interpolation of Lebesgue spaces $L^{p_0}$ and $L^{p_1}$ are well-known, that is,
$[L^{p_0}, L^{p_1}]_\theta = L^p$, where $(1-\theta)/p_0+1/p_1 = 1/p$ for some $\theta \in [0,1]$.
Now I ...
- 193