All Questions
1,304 questions
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:
$$\Vert u(t,\cdot)\Vert_{L^...
- 303
8
votes
3
answers
884
views
abstract evolution equations
Hi
Whenever I read a book on evolution equations, they set up, say the parabolic PDE
$$\dot{y} = Ay + f$$
in abstract function spaces (eg. $L^2(0,T;V)$ and $L^2(0,T;V^*)$). In examples, they always ...
- 52.3k
1
vote
1
answer
393
views
Decay estimates for solutions to the damped wave equation
Consider the following Cauchy problem:
$$u_{tt} + u_t - \Delta u = 0,$$
$$u(0,x)=u_0(x) \in L^1 \cap L^2,$$
$$u_t(0,x) = u_1(x) \in L^1 \cap L^2.$$
Question: By means of Fourier transform and ...
- 39.1k
2
votes
0
answers
90
views
Boundary regularity of solutions to semilinear heat equation
Consider the Cauchy IVP problem
$$u_t - \Delta u + f(t,x,u) = 0, \quad (t,x) \in (0,T) \times \mathbb{R}^n,$$
with initial condition $u(0,x) = g(x), \ x \in \mathbb{R}^n.$
Can you point out a ...
- 303
2
votes
1
answer
1k
views
Pointwise convergence implies uniform convergence?
Let $K$ be an integral kernel of a bounded operator $S:L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n) $ defined like
$$(Sf)(x)= \int_{\mathbb{R}^n}K(x,y)f(y)dy.$$
Assume that $K\in C^{\text{bounded}...
- 24.2k
10
votes
1
answer
3k
views
Trace of integral trace-class operator
I have seen many answers to the converse question (which seems to be difficult in general), but I would like to ask the following:
Let $T: L^2 \rightarrow L^2$ be a trace-class operator that is also ...
- 109k
7
votes
1
answer
489
views
When the value of a function in a point is equal to its integral average over the point's neighborhood?
It is well-known that the harmonic functions have this remarkable Averaging Property: if $f$ is harmonic in a domain $U \subset R^n$, then, for any point $x \in U$, $f(x)$ is equal to the integral ...
- 91
1
vote
0
answers
76
views
Which sets support which spectra?
I know (and this is of course rather elementary) that an isolated point in the spectrum of a self-adjoint operator $T$ always belongs to the point-spectrum.
I would like to ask: Are there similar ...
- 173
3
votes
1
answer
415
views
Continuity of convolution on L¹
Question
Let $f,g\in L^1(\mathbb{R}^n)$ be nonnegative and assume $\int f\,g \; dx <\infty$. Let $\Phi_\varepsilon$ be the heat kernel at time $\varepsilon$. Denote with $f_\varepsilon = f \ast \...
- 23.2k
0
votes
0
answers
81
views
Differential operator and equivalence
Here is the problem:
I have a certain PDE and there is the nonlinear terme $h$, I have as data:
$f \in H_0^2(0,L)$,,,$g \in {H^1}(0,L)$ with ${g_x}(0) = {g_x}(L) = 0$
Now on consider the fnction $$h(...
- 617
0
votes
1
answer
152
views
Solution of Poisson equation vanishing at the boundary of any order
Let $f$ be a compactly supported function in $\Omega \subset \mathbb{R}^3$ and
$\Delta u=f$ in $\Omega$
such that $D^{\alpha}u=0$ on $\partial \Omega$ for every multi-index $\alpha$ with $|\alpha| \...
- 91.8k
6
votes
2
answers
530
views
Schrödinger eigenfunctions are bounded
Let $V:\mathbb{R}\rightarrow \mathbb{R}^{+ *}$ a real positive function such that $\displaystyle \lim_{ x \to \pm\infty} V(x)= +\infty $.
Then the Schrödinger operator $H=-\frac{d^2}{dx^2}+V(x)$ has ...
- 8,336
3
votes
1
answer
273
views
References on nonlinear evolution equations treated as infinite-dimensional systems for nonexperts
In many cases of interest a nonlinear evolution partial differential equation can be written as an infinite-dimensional dynamical system
$$
du/dt+A(t)u=0
$$
on a suitable functional space $X$, where $...
CommunityBot
- 1
4
votes
1
answer
173
views
Compactness modulo symmetries of critical NLS solution
I was referred to the paper C. Kenig and F. Merle, "Global well-posedness, scattering, and blow-up for the energy critical, focusing non-linear Schrodinger equation in the radial case", in which one ...
- 873
2
votes
0
answers
127
views
slightly subcritical elliptic pde; the linearized equations
Let $ p_m \nearrow \frac{N+2}{N-2}$ and consider the family of elliptic problems
$$-\Delta u_m(x)=u_m(x)^{p_m} \quad B \qquad \quad u_m =0 \quad \partial B,$$ where $B$ is the unit ball ...
CommunityBot
- 1
3
votes
1
answer
146
views
Radial Kernel with Bounded Support and Norm of Gradient Bounded by a Dimension-free Constant
I was wondering if it is possible to construct a compactly supported radial kernel function in $\mathbb{R}^d$ such that the norm of the gradient is bounded by some dimension-free constant. That is, ...
CommunityBot
- 1
2
votes
0
answers
218
views
Existence of solutions to time-dependent Schrödinger equations
I would like to know what is known about evolution equations of the form
$$iy'(t)=H_0y(x,t)+u(t)V(x)y(x,t)$$
and $y(0)=y_0 \in D(H_0)$
where
$V$ is not a bounded operator, but an unbounded one, $u \...
- 6,881
3
votes
0
answers
210
views
Meromorphic continuation of resolvent of free Laplacian on homogeneous Sobolev space
Let $n \ge 2$. Set $\dot{H}^1(\mathbb{R}^n)$ to be the homogeneous Sobolev space, defined as the Hilbert completion of $C_0^\infty(\mathbb{R}^n)$ with respect to the norm $\| \varphi \|^2_{\dot{H}^1} \...
- 481
4
votes
1
answer
134
views
A nice function space closed with the operation $x \cdot \nabla $
I am studying a certain model kinetic equation. To study that system, I have to find a function space, which is a subspace of $L^1 (\mathbb{R}^d)$ and the operator $f \rightarrow x \cdot \nabla_x f$, ...
- 16.5k
4
votes
2
answers
821
views
Elliptic regularity in $L^1$
Dear all,
I am looking for a good reference for elliptic regularity in $L^1$. To be more precise
Let $\Omega\subset\mathrm{R}^n$ be a bounded smooth domain, let $A$ be a properly elliptic ...
0
votes
0
answers
308
views
Invertible operator
We consider the operator $$T=I + {{{\partial ^2}} \over {\partial {x^2}}}:{H^2}(0,L) \cap H_0^1(0,L) \to {L^2}(0,L)$$
We hope to prove that $T$ is invertible if and only if $L = n\pi $.
and for this ...
- 6,259
1
vote
0
answers
183
views
Trace theorem for boundary value problem
Consider the inhomogeneous boundary value problem on the infinite strip $(x,y)\in \mathbb{R}\times [0,1]$ defined by
$$\begin{cases}\partial_{x}u + \partial_{y}v=f & {(x,y)\in \mathbb{R}\times (0,...
- 873
4
votes
0
answers
89
views
How can I can derive an explicit bound for the solution of the poisson's PDE?
i need some help on this question
Let $\Omega$ be an open subset of $\mathbb{R}^{2}$ (say a square) with
$\partial{\Omega} =\Gamma_{1} \cup \Gamma_{2} \cup\Gamma_{3} \cup\Gamma_{4}$. A structure ...
- 325
11
votes
3
answers
3k
views
Dual space of $L^2(\mathbb{R},L^1(0,1))$?
I was wondering what the dual space of $L^2(\mathbb{R},L^1(0,1))$ is? (equipped with Lebesgue measures)
Formally, one would suspect that it is just $L^2(\mathbb{R},L^{\infty}(0,1))$. But this may be a ...
CommunityBot
- 1
1
vote
1
answer
130
views
Resolvent difference of absolute values!
Let $T$ be a bounded operator. Then, the operators $\left\lvert T \right\rvert:=\sqrt{T^*T}$ and $\left\lvert T^* \right\rvert:=\sqrt{TT^*}$ are well-defined.
Is there a way to write
$$(\left\lvert ...
CommunityBot
- 1
2
votes
1
answer
102
views
Evolution equation invariance of sets
Let $A: D(A) \subset X \rightarrow X$ be a generator of a $C_0-$semigroup and $Z$ be a bounded operator on $X$, then the evolution equation for $u \in C([0,T], \mathbb{R})$
$$\varphi'(t) = A \varphi(t)...
- 16.5k
2
votes
0
answers
226
views
degree theory argument in elliptic pde; apparent contradiction
i have a question regarding a degree theory argument and an apparent contradiction. Let me point out that I am a complete novice with degree theory and really i am just pushing some symbols with no ...
- 1,385
1
vote
0
answers
180
views
Implicit function theorem for operators
Let $P: (-a,a) \rightarrow \Psi_h^0(\mathbb{R}),$ be a pseudodifferential operator in Weyl quantization with $(-a,a) \ni z \mapsto P(z)$ depending smoothly on this parameter $z$. Note that this ...
CommunityBot
- 1
3
votes
0
answers
280
views
Helmholtz-Hodge decomposition
I have a question regarding a decomposition of a vector field. So fix $ 1<p<\infty$ and let $ \Omega$ denote a smooth bounded domain in $ R^N$. Now let $ F $ denote a smooth vector field $F:\...
- 1,385
5
votes
1
answer
1k
views
Trace-norm of integral operator
Let me start by saying that I do appreciate any insight on this. So also if you have a partial result, please share it as a comment or answer.
This is somewhat unrelated to what I normally do, so I ...
- 24.2k
0
votes
1
answer
104
views
Operator identity for convergent series
Let $T_i$ and $S_i$ be a sequence of bounded operators such that
$$ \sum_{k,i,j=0}^{\infty} S_j^* T_i^* T_i S_k$$ converges unconditionally in operator norm on some Hilbert space. The limit is then ...
- 23
1
vote
0
answers
124
views
Singular value decomposition in two spaces (reference in Russian paper?)
Let $H$ be a Hilbert space and $X$ be a Banach space such that $H \cap X$ is dense in both.
Now, let $T$ be an operator such that $T: H \rightarrow H$ and $T:X \rightarrow X$ exists in the sense that ...
2
votes
1
answer
147
views
Showing existence of minimisers with single integral constraint on a possibly non-Lipschitz domain?
Consider a domain $\mathcal{D}$ which is a right circular cylinder in $\mathbb{R}^3$, with radius 1 and height 1, say. The boundary of $\mathcal{D}$, which I denote by $\partial\mathcal{D}$ consists ...
CommunityBot
- 1
2
votes
2
answers
141
views
Equality of spectra of products of operators
Let $A$ be a linear operator between two Hilbert spaces. Let $A^*$ be its adjoint.
Question. Under what conditions the non-zero spectra of $A^*A$ and $AA^*$ coincide counting multiplicities?
In my ...
- 658
2
votes
0
answers
115
views
Does this Sobolev-space like construction have a name?
Take $\Omega \subset \mathbb{R}^n$ arbitrary then define as $X$ the closure of $C^1(\Omega) \cap W^{1,1}(\Omega)$ w.r.t. the norm $f \mapsto \left\lVert f \right\rVert_{\infty} + \left\lVert \nabla f \...
- 305
1
vote
1
answer
162
views
Regularity of integral kernel
Let $\Omega \subset \mathbb{R}^n$ be some open set.
If, for all $\psi \in L^2(\Omega)$ and some fixed integral kernel $k \in L^2(\Omega\times \Omega)$ and $\ell>0$, it is true that both
$\int_{\...
- 1,191
3
votes
1
answer
247
views
Are there fundamental solutions of the laplacian that decay rapidly?
The question
I consider the Laplacian $\Delta = \partial_1^2 + \partial_2^2 + \partial_3^2$ in $\mathbb{R}^3$. By the "standard" fundamental solution of the Laplacian, I mean the function
$$ \...
- 24.2k
1
vote
1
answer
518
views
Interpolation between Schatten classes
I was wondering if there is an analogue to the classical Riesz Thorin theorem for Schatten classes. I suppose the answer is yes, since Schatten classes are so similar to $\ell^p$ spaces for which the ...
0
votes
2
answers
197
views
Concerning the decay of the ground state of certain Schrodinger operators
Consider the Schrodinger operator in $n$ dimensions with a potential $V$, which grows rather quickly as $\mid x\mid$ tends to infinity, but with negative potential in a bounded region, for example, a ...
- 219
4
votes
1
answer
165
views
Scattering of relativistic particle by long-range potential
Let
$\mathcal{H}=L^2(\mathbb{R}^3)$,
$H_0=\sqrt{-\Delta+M^2}$, ($M$ is a positive constant, $\Delta$ is the laplacian)
and
$H=H_0+V(\vec{x})$
(where $V(\vec{x})$ is the operator of ...
- 1,191
1
vote
3
answers
192
views
An acting condition for a superposition operator from $H^1(\Omega)$ to $H^1(\Omega)$
If i take $v\in H^1(\Omega)$ where
$$
H^1(\Omega)=\{u\in L^2(\Omega), \frac{\partial u}{\partial x_i}\in L^2(\Omega), i=1,\ldots,N\}
$$
$\Omega$ is bounded open set from $\mathbb{R}^N$
What is the ...
- 446
2
votes
0
answers
235
views
The Cauchy problem associated with $u_t^\epsilon + H(x,t,u^\epsilon,\nabla u^\epsilon) = \epsilon\Delta u^\epsilon$
Consider the initial value problem $$\begin{cases} u_t^\epsilon + H(x,t,u^\epsilon,\nabla_x u^\epsilon) = \epsilon\Delta_x u^\epsilon & \text{ in } \mathbb{R}^n \times (0,\infty)\\ u^\epsilon = g &...
CommunityBot
- 1
2
votes
1
answer
699
views
Schwartz kernel theorem
I would like to understand how the Schwartz kernel theorem works for some more difficult cases and therefore would like to discuss an example from scratch:
Let the Dirichlet Laplacian on the half-...
- 1,191
0
votes
1
answer
129
views
Composition of a negative operator and a positive one
Let $\Omega$ a bounded open set of $R^n$, $\omega$ a non-empty open subset of $\Omega$ and
$\chi_{\omega} : L^2(\Omega) \longrightarrow L^2(\omega)$
be the restriction operator to $\omega$,...
- 42.8k
2
votes
0
answers
140
views
Question on the differentiability of the solution mapping in the obstacle problem
I'm looking for a reference for the following. Take the obstacle problem:
$$\int_\Omega \nabla u \nabla (v-u) \leq \int_\Omega f(v-u)$$
for a function $u \in K:=\{v \in H^1_0(\Omega) : v \geq \varphi\}...
- 2,670
2
votes
1
answer
449
views
Derivative of Yosida-Approximation
i have got a problem with some assumptions to solve a parabolic variational inequality. My Problem is: Find a function $u$ with
\begin{align}
u\in L^2(0,T;V),~ u' \in L^2(0,T;V') \\
(u'(t),v-u(t)) + ...
- 1,587
1
vote
1
answer
326
views
Weyl sequence for $H_a=\frac{d^2}{dx^2}+a^2x^2$ [closed]
We consider the following Operator: $H_a=\frac{d^2}{dx^2}+a^2x^2$ were $a\in R^{*}$.
Let be $b\in R$. I want to construct a sequence $w_n$ (Which depends on $b$)such that:
1-$||w_n||_2=1$.
2-$w_n$ ...
CommunityBot
- 1
1
vote
0
answers
50
views
Verifying general assumption for parabolic PDE
I've got some problems verifying an assumption for a parabolic PDE. Namely, let $(V,H,V^*)$ be a Gelfand-Triple, $u_0 \in V$, $\psi\colon V \to \mathbb{R}$ convex and lower-semicontinuous and $a\colon ...
- 187
1
vote
0
answers
664
views
$W^{2,p}$ regularity of elliptic PDEs with Neumann boundary condition
Given an elliptic PDE with Neumann boundary condition
\begin{align}
\left\{
\begin{aligned}
-\sum_{i,j=1}^N\partial_i(a_{ij}\partial_j u)+cu&=f &&\mbox{in}\,\,\,\Omega, \\
\sum_{i,j=1}^Na_{...
CommunityBot
- 1
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