All Questions
Tagged with ap.analysis-of-pdes fa.functional-analysis
1,304 questions
5
votes
2
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Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $
I have a question that I could not find it any where.
Is the completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm
$$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \...
- 371
5
votes
1
answer
351
views
Does the Poincaré inequality hold on annular domains?
Does the following Poincaré inequality hold
$$\int_{B_{r_2}\setminus B_{r_1}} |f-\bar{f}|^2 dx \leq C(r_2-r_1)^2 \int_{B_{r_2}\setminus B_{r_1}} |\nabla f|^2 dx,$$
where $B_r$ denotes a ball of radius ...
- 537
5
votes
3
answers
1k
views
Decay estimate for the heat equation: $\sup_{t>0}\int_{\mathbb{R}} t^\alpha |u_x|^2\ dx$
Let $u$ be a solution of the heat equation $$u_t - u_{xx} = 0, \quad t>0, x \in \mathbb{R}$$
with initial data $u(0,\cdot) = u_0$.
Fix $\alpha >0$. How can I estimate (without using explicitly ...
- 839
5
votes
2
answers
358
views
Linear transport equation with unbounded coefficients
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = f_0(x) \in \mathscr S(\mathbb R^n).$
I am wondering then if $q$ and all its ...
- 124
5
votes
2
answers
976
views
Symbol of the Laplace-Beltrami on $\mathbb{S}^2$
This question is about how the principal part (or symbol) is defined on a manifold?-I assume that the answer is: As in $\mathbb{R}^n$ using local coordinates, i.e.
A differential operator $P=\sum_{|\...
- 329
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
5
votes
2
answers
936
views
Second order differentiability of subharmonic function almost everywhere?
The following general definition of subharmonic function comes from the classical text book [elliptic partial differential equations of second order] by Gilbarg and Trudinger.
We call a function $u$ ...
- 1,350
5
votes
2
answers
699
views
Ground state for non-linear Schrödinger
When studying the blow-up for focusing non-linear Schrödinger equation (NLS) one often compares the initial-state to a stationary solution.
In the energy-critical case, this stationary solution is ...
- 536
5
votes
1
answer
928
views
Do exist infinitely differentiable, compactly supported non zero solutions of the free Schrodinger equation?
I would like to get an answer for the following problem (and possibly be pointed to the relevant literature): given the one dimensional free Schrodinger equation $ i \, f_t + f_{xx}/2 = 0$ for the ...
- 53
5
votes
1
answer
932
views
Mellin transform between heat kernel and zeta-function
For some notion of a "positive operator" $D$ of "Laplacian type" one seems to be able to define a notion of a zeta-function as $\xi(s,f,D) = Tr_{L^2}(f D^{-s})$ where $f \in L^2$ (the space of square-...
- 617
5
votes
3
answers
490
views
Continuity with values in L^2
Hi,
let $T>0$, $\Omega\subset\mathrm{R}^n$ be a bounded smooth domain and suppose
$$u\in L^2(0,T;W^{1,2}(\Omega))\cap L^\infty((0,T)\times\Omega))\ \text{and } \partial_tu\in L^2(0,T;W^{-1,2}(\...
- 103
5
votes
2
answers
459
views
Backward heat equation and forward perturbed heat equation well posed?
I consider the following scenario. Let $I$ be a compact interval in space and $f$ a nice function in the space $C^{\infty}(I)$. In the following we consider a self-adjoint realization of our operators ...
- 536
5
votes
3
answers
1k
views
Showing $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?
Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain.
How does one prove that the inclusion $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?
I define $H^{\frac 1 2}(\...
- 307
5
votes
2
answers
364
views
Euler-Lagrange equations for minimizer of energy with indicator function
I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for
$$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
- 93
5
votes
1
answer
487
views
Fokker-Planck: equivalence between linear spectral gap and nonlinear displacement convexity?
In a smooth, bounded and convex domain $\Omega\subset \mathbb R^d$ consider the usual linear Fokker-Planck equation with Neumann (some would say Robin) boundary conditions
\begin{equation}
\label{FP}
\...
- 5,380
5
votes
2
answers
285
views
Existence of a solution for this hypoelliptic-alike PDE
I know that this question may result rater vague and somehow out of context, still I am hoping you could help me.
Assume we have the following equation
\begin{align}
\boxed{\partial_t u(t,x,z)=\...
- 515
5
votes
2
answers
233
views
Analytic approximations of smooth vector fields
Let $M$ be the set of smooth divergence-free vector fields $u$ on $\mathbb{R}^3$ with
$$|\partial_x^{\alpha} u(x)| \leq C_{\alpha K}(1+|x|)^{-K}$$
on $\mathbb{R}^3$ for any $\alpha,K$.
Further, we ...
- 749
5
votes
1
answer
216
views
Bounds on dimension of a subspace
Let $I=(0,1)$ and let $C>1$ be a constant. Let $L^2(I)$ and $H^1(I)$ be the standard Sobolev spaces on $I$. Suppose that $U$ is a subspace of $H^1(I)$ with the additional property that:
$$ \| u\|_{...
- 4,135
5
votes
1
answer
224
views
Spectral theory of infinite volume hyperbolic manifolds
I have a question about the discrete spectrum of the Laplace operator on hyperbolic manifolds with infinite volume. I understand the case of infinite area surfaces: see Chapter 7 (Sections 1 and 2) of ...
- 1,407
5
votes
1
answer
743
views
Eigenvalues and Domain of the Laplace-Beltrami Operator
Assume $(M,g)$ is a compact Riemannian manifold without boundary, where $g$ is the Riemannian metric. Let $L:=-\Delta$ be the Laplace-Beltrami operator on $M$ defined by $\Delta \cdot = \text{div}(\...
- 51
5
votes
1
answer
395
views
Universal decay rate of the Fisher information along the heat flow
I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation
$$
\partial_t u=\Delta u
$$
...
- 5,380
5
votes
1
answer
1k
views
$L^\infty$ estimate on heat equation with a lower order term
Let $u$ be the weak solution on a smooth bounded domain $\Omega \subset \mathbb{R}^n$ (for $n \leq 3$) of
$$u_t - \Delta u = f$$
$$u(0) = u_0$$
$$\partial_\nu u = 0 \quad\text{on $\partial\Omega$}$$
...
- 53
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 $H^1_0(\Omega)$...
user14319
5
votes
1
answer
1k
views
Are Besov spaces $B^{s}_{p,q}$ invariant under Fourier transform?
(This may be very easy question for MO; as I am just trying to understand Besov spaces)
Let $\phi \in C^{\infty}(\mathbb R^{n})$ with
$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|...
- 1,051
5
votes
1
answer
660
views
Hormander's bracket condition for the adjoint of an operator
Let $X_0, X_1, \dots, X_k$ be smooth vector fields over ${\mathbb R}^n$, and let us consider the operator
$$
L = \sum_{i=1}^k X_i^2 + X_0~.
$$
Here, I assume that Hörmander's bracket condition is ...
- 135
5
votes
1
answer
512
views
$C_0$-semigroups applications
My graduation thesis was about stability theorems for $C_0$-semigroups (see the Wikipedia article for the definitions: http://en.wikipedia.org/wiki/C0-semigroup). I would like to know if there is ...
- 2,222
5
votes
3
answers
2k
views
Characterizing the harmonic oscillator creation and annihilation operators in a rotationally invariant way
I am interested in a characterization of the creation and annihilation operators that is in some sense invariant under $O(n)$ rotations of $\mathbb{R}^n$:
Background
The Harmonic Oscillator on $\...
- 7,197
5
votes
1
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453
views
Seeking for references on some PDEs
This is not a technical mathematical question. I came across some PDEs with no references nor their names.
$$-\Delta u + \int_\Omega udx = f\qquad \hbox{in $\Omega$} \label{1}\tag{Eq1}$$
The above ...
- 1,101
5
votes
1
answer
260
views
Approximate Sobolev embedding
It is well-known in $H^2(\mathbb R^3)$ embeds into $L^{\infty}(\mathbb{R}^3).$ Now consider a function $u \in \ell^{\infty}(h\mathbb Z^3)$ and a grid of points $x \in h\mathbb{Z}^3.$
We then define ...
- 124
5
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1
answer
571
views
Schrödinger operator with Coulomb potential
The free Laplacian $-\Delta$ has absolutely continuous spectrum $[0,\infty).$ The Coulomb Hamiltonian $H=-\Delta-\frac{1}{\vert x\vert}$ on $L^2(\mathbb R^3)$ has absolutely continuous spectrum $[0,\...
- 119
5
votes
1
answer
171
views
Invariant subspace in infinite dimensions
Let $A(t)$ be a family of skew self-adjoint operator defined on some Hilbert space $H$ with common domain $D(A).$
The dependence on $t$ is in the strongly continuous sense, i.e. for all $x \in D(A)$ ...
- 177
5
votes
1
answer
471
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
$$
W^{s_1,s_2}_{p}:=\left\{f: ...
- 515
5
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1
answer
421
views
application of factorization theorem
Young's inequlity tells us that $L^{1}(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R)$ with norm inequality
$$\|f\ast g\|_{L^{p}} \leq \|f\|_{L^1}\|g\|_{L^p};$$
and of course this ...
- 1,051
5
votes
1
answer
764
views
Weak convergence in $W^{1,p}_0$
Note from the answerer : this question stems from this article.
I ask this question in https://math.stackexchange.com/questions/1206617
I have a bounded sequence $(u_n)$ from $W^{1,p}_0(\Omega)$ so ...
- 277
5
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2
answers
1k
views
Compactly supported functions and Sobolev spaces on manifolds
It is well-known that if a complete Riemannian manifold has bounded curvature and injectivity radius bounded away from zero, then the space $C^\infty_c(M)$ is dense in the Sobolev spaces $W^{k, p}(M)$ ...
- 9,853
5
votes
1
answer
246
views
An asymmetric quadrilinear estimate
Fix $1<p<2$ and let $a_{i}=1-\frac{\theta_{i}}{p^{\prime}}$
where $\theta_{i}\in (0,1/2)$, $i=1,2,3,4$, and $p^{\prime}$ is the conjugate exponent of $p$. Note here that $0<a_{i}=1-\theta_{i}+...
- 852
5
votes
1
answer
170
views
Regular Lagrangian flow for "square root example": $\frac{d}{dt} X(t,x) = \sqrt{X(t,x)}$
Consider the problem
$$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \sqrt{X(t,x)}, &t \in [0,T],\\
X(0,x) = x, &x \in \mathbb R
\end{cases}
$$
This is the prototype of non-uniqueness ...
- 839
5
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1
answer
630
views
Uniqueness of Kantorovich potentials?
$\newcommand{\R}{\mathbb R}$Take $\Omega\subset \R^d$ bounded, convex, and smooth.
Consider the optimal transport problem with cost $c(x,y)=\lvert x-y\rvert^2$, leading to the quadratic Wassersein ...
- 5,380
5
votes
1
answer
281
views
de Rham theorem for tempered distributions
I am wondering if the following statement holds.
If $u\in \mathscr{S}'$ satisfies $\left< u,\Phi\right>=0$ for all $\Phi \in \mathscr{S}$ with $\mathrm{div}\, \Phi=0$, then there exists $p\in \...
- 323
5
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1
answer
564
views
Convergence of discrete Laplacian to continuous one
I make the following observation:
Let $\Delta^{(n)}$ be the discrete Laplacian on $\mathbb{C}^n$ (ie the $n\times n $ matrix with diagonal $-2$ and upper/lower diagonal $1$.)
This one has eigenvalues ...
- 536
5
votes
1
answer
153
views
Why is density and separability needed for uniqueness of weak (time) derivatives?
Let $X,Y$ be Banach spaces with $X \subset Y$. Recall that $u \in L^1(0,T;X)$ has weak derivative $g \in L^1(0,T;Y)$ if
$$\int_0^T u(t)\phi'(t) = -\int_0^T g(t)\phi(t) \qquad\forall \phi \in C_c^\...
5
votes
1
answer
211
views
Pointwise convergence in functional calculus
Let $A_n$ be a family of (bounded) self-adjoint operator converging pointwise to some (unbounded) self-adjoint operator $A,$ i.e. for all $x$ in the domain of $A$
$$\left\lVert A_n x-Ax \right\rVert \...
- 65
5
votes
1
answer
379
views
References on the obstacle problem for the heat equation
Can you point out some references that deal with the obstacle problem for the heat equation?
$$(OP) \quad\begin{cases}
\max\{\Delta u -\partial_t u, \varphi - u \} = 0 & \text{ in } (0,T)\times \...
user108310
5
votes
1
answer
263
views
Characterization of the interpolation space $(X,D(A^\alpha))_{\theta,p}$ with semigroup $A$ generates?
Let $X$ be a Banach space (could work for over $\mathbb{R}$ as well?)
Let $A\colon D(A)\subset X\to X$ be a sectorial operator, and $e^{tA}$ be the semigroup generated by $A$.
It is well-known that ...
- 153
5
votes
1
answer
2k
views
About Aubin-Lions Lemma
I have a question about Aubin-Lions Lemma, the standard Aubin-Lions lemma need those Banach Space be reflexive spaces, are there any version of Aubin-Lions without reflexivity?
Standard aubin-lions:...
- 103
5
votes
1
answer
311
views
Maximal operator estimates for the Schrödinger equation
Let $a>0$ and consider the operator
$$Tf(t,x)= \int_{\mathbb{R}^{n}}e^{ i x\cdot \xi} e^{i t \lvert\xi\rvert^{a}} \widehat{f}(\xi) \, d\xi.$$
When $a=2$, the function $Tf$ solves the Cauchy problem ...
- 852
5
votes
1
answer
279
views
Connecting PDE notions for functions $[0,T] \to (\Omega \to \mathbb{R})$ to related notions for functions $[0,T] \times \Omega \to \mathbb{R}$
Fix $\Omega \subseteq \mathbb{R}^N$ a bounded domain (of whatever smoothness you end up needing, let's say $C^1$ domain for definiteness) and fix some $0 <T < \infty$. In considering evolution ...
- 324
5
votes
2
answers
679
views
Banach algebra of smooth functions
To fix the ideas, let's work on the flat periodic torus $\mathbf{T}^d:=\mathbf{R}^d/\mathbf{Z}^d$.
My question is the following.
Does there exist an infinite dimensional Banach (sub-)algebra $A \...
- 3,425
5
votes
1
answer
448
views
improved Sobolev embedding
This is probably not a research level question but I am struggling with the geometry. My question is related to whether some monotonicity can increase the range of exponents in the Sobolev embedding.
...
- 1,385
5
votes
1
answer
361
views
Exponential decay of solution in $L^p$ with $p>2$
Consider the following evolution equation
$$u_t=\Delta u$$
in a bounded and regular open subset $\Omega$ of $\mathbb{R}^N$, with smooth initial conditions $u_0\geq 0$ and homogeneous Dirichlet ...
- 161