All Questions
Tagged with fa.functional-analysis ap.analysis-of-pdes
1,304 questions
8
votes
1
answer
5k
views
integration by parts for the fractional Laplacian
Is there an integration by parts formula for fractional laplacians in $L^p(\mathbb{R}^N)$, something like
$$
s\in(0,1),\qquad\int\limits_{\mathbb{R}^N}f[(-\Delta)^sg] =\int\limits_{\mathbb{R}^...
- 5,380
3
votes
1
answer
661
views
What is visualization of gradient flow of a functional?
I don't work on functional analysis but during my study, I faced gradient of a functional. I read its definition, but I can not understand why it is a useful tool? Why if a flow can be written as a ...
- 1,303
3
votes
1
answer
733
views
Trace theorem for manifolds with boundary
Can I get a reference to a trace theorem for a manifold $M$ with boundary $\partial M$, and I am hoping the inequality
$$\lVert Tu \rVert_{L^2(\partial M)} \leq c\lVert u \rVert_{H^1(M)}$$
will hold.
...
1
vote
1
answer
367
views
weak*closure of {f:||f||=1} in dual.
What is the weak* closure of {f:||f||=1}? I am sure this set is not closed in weak* topology.
So what is the weak* closure of this set. Thanks.
- 1,269
0
votes
0
answers
166
views
Harnack's Inequality and (hypo)elliptic PDE
Background: I am aware of the Harnack's Inequality for linear elliptic equations.
My questions are:
(a) Is there a version of Harnack's Inequality for nonlinear elliptic equations, say, of the form ...
- 1
4
votes
3
answers
1k
views
Using Galerkin method for PDE with Neumann boundary condition?
I am wanting to show existence of solutions to
$$u_t +L(u) = f \;\;\text{on}\;\; \Omega$$
with initial condition $u|_{t=0} = u_0$ and Neumann boundary condition $\nabla u\cdot \nu = 0$ on ${\partial\...
- 113
3
votes
1
answer
1k
views
Showing a singular integral operator takes Hölder continuous functions to Hölder continuous functions of the same order
I would like to show the following function is $\gamma$-Hölder continuous. Said function $F:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined by a singular integral operator of convolution type as ...
- 31
1
vote
0
answers
294
views
Galerkin method for existence for PDE with nonsymmetric bilinear form
Suppose we have a PDE
$$\langle u', v \rangle + a(u,v) = 0$$
where $a:V\times V \to \mathbb{R}$ is a bounded symmetric bilinear form, then if $u_0 \in V$ then $u \in L^2(0,T; V)$ with $u' \in L^2(0,T;...
- 113
4
votes
1
answer
367
views
Manifold structure for the set of solutions to a first order elliptic system?
Consider a bounded domain $S\subset R^2$ and an elliptic system of two first order PDEs, namely a generalization of the Cauchy-Riemann system allowing nonconstant coefficients and lower order terms. ...
- 41
10
votes
1
answer
957
views
Do eigenfunctions of elliptic operator form basis of $H^k(M)$?
We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$ and $L^2(M)$.
If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ ...
- 103
3
votes
0
answers
381
views
Extension divergence-free, curl-converging vector field
Hi.
Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole ...
- 3,425
0
votes
0
answers
214
views
Splitting the action of functionals in duals of Sobolev spaces
Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
- 101
3
votes
1
answer
785
views
on an inequality of Brezis-Lieb
In their 1983 JFA paper Brezis and Lieb have shown, among many other things, a Poincaré-type inequality: in the case of a harmonic function $f$ on a bounded domain $\Omega$, their inequality ((3.14) ...
- 3,388
1
vote
0
answers
315
views
"Integration by parts" formula for functionals
We know that for a Hilbert triple $V \subset H \subset V^{'}$, if we have $u, v \in L^2(0,T;V)$ with $u',v' \in L^2(0,T;V')$
then
$$\frac{d}{dt}(u(t), v(t))_H = u'(t)(v(t)) + v'(t)(u(t))$$
where the $...
- 29
3
votes
1
answer
2k
views
Norm of differential operator between Sobolev spaces
It is easy to check that the differential operator $\partial^a$ (where $\alpha\in \mathbb{N}_0^n$) is continuous between the Sobolev spaces (with usual norms)
$W^{m,p}(U)\to W^{m-|\alpha|,p}(U)$, ...
- 31
7
votes
2
answers
2k
views
Uniform bound on the eigenfunctions of the Laplacian
Is it possibly to have $L_\infty$ bounds on the eigenfunctions of the Laplacian operator on bounded regular domains with Dirichlet condition? I found several papers by Sogge but these are pretty ...
- 73
5
votes
1
answer
418
views
Robin-Laplacian in unbounded domains
Let $\Omega\subset \mathbb R^n$ be an open domain and $\tau>0$. Consider the following boundary value problem
$-\Delta v=f $ in $\Omega$, $\partial_\nu v+\tau v=g$ on $\partial\Omega$.
If $\Omega$...
3
votes
1
answer
354
views
Solvability for constant-coefficient partial differential operators
Let $\mathcal{S}$ denote the space of Schwartz functions on $\mathbb{R}^n$, and $\mathcal{S}'$ the space of tempered distributions. Let $L$ denote a linear, constant-coefficient, partial differential ...
- 111
3
votes
1
answer
747
views
Reference request: Anisotropic Sobolev spaces
Hello,
I am interested in what is known about anisotropic Sobolev spaces, by which I mean spaces of functions satisfying
$ \| f \|_p < \infty, \|Df \|_q < \infty, $
where $p \ne q$ (as ...
- 411
11
votes
3
answers
1k
views
Boundedness of the derivative of the trace of an H^1 function
As a research preface, this question is linked to a problem of increasing magnetism in Ginzburg-Landau equations that I have distilled for the purpose of getting to the bottom of this technical matter....
- 882
1
vote
0
answers
205
views
Looking for higher order Sobolev inequality
Hello,
On a compact (without boundary) Riemannian manifold (eg. some surface in $\mathbb{R}^n$), I'm looking for a result like
$$\lVert \nabla u\rVert_{L^2}^2 \leq \epsilon\lVert u\rVert_{H^1}^2+\...
- 29
1
vote
1
answer
479
views
Existence of solution for this parabolic PDE
The parabolic PDE
$$\langle u', v \rangle + a(u,v) = \langle f, v \rangle$$
has a unique solution $u \in L^2(0,T; H^1)$ with $u' \in L^2(0,T;H^{-1})$ if $a$ is a bounded and coercive bilinear form (...
- 107
2
votes
2
answers
326
views
Are $\lVert \Delta u \rVert_{L^2(S)}$ and $\lVert u \rVert_{H^2(S)}$ equivalent norms on a compact manifold?
Hi,
I am looking for the result:
$$\text{The norm} \quad \lVert \Delta u \rVert_{L^2(S)} \quad \text{is equivalent to} \quad \lVert u \rVert_{H^2(S)}$$
for scalar functions $u \in H^2(S)$, where $S$ ...
- 55
1
vote
1
answer
1k
views
weak derivative and continuous function
Let $\Omega \subset \mathbb{R}^n$ be a compact smooth hypersurface. Suppose $\varphi \in C_c^\infty(0,T; H^1(\Omega))$ is a $H^1(\Omega)$-valued test function (so $\varphi(t) \in H^1(\Omega)$ for each ...
- 107
13
votes
3
answers
2k
views
Space of sections of a fibre bundle with non-compact base space
Let $\pi: E \rightarrow M$ be a fiber bundle over the manifold M and denote by $\Gamma(E)$ the space of smooth sections of $E$.
For compact $M$ it is well known (Hamilton 1982, Part II Corollary 1.3.9)...
- 5,824
1
vote
2
answers
847
views
Fourier transform of function on compact set and Sobolev norm equivalence
Hi all. My question on M.SE is unanswered (https://math.stackexchange.com/questions/254970/fourier-transform-of-function-defined-on-subset-of-mathbbrn) so I want to post it here. I changed it slightly....
- 113
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
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
0
votes
1
answer
1k
views
Showing a coercivity condition for this bilinear form
Suppose $\Omega \subset \mathbb{R}^n$ is a compact domain. Let $f$ and $J$ (and also $\frac 1J$) be $C^1$ functions on $\Omega$. Consider the bilinear form $a:H^1(\Omega) \times H^1(\Omega) \to \...
- 107
1
vote
2
answers
938
views
Alternate definitions of $C^{1,\alpha}$ and $C^{1,\alpha}(\bar{D})$ maps
My question is about the precise definition regarding the following:
Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a ...
- 1,471
1
vote
2
answers
522
views
Variational problems whose lagrangian density depends on derivatives higher than 1.
The usual theory of calculus of variations, as far as I know, is concerned with lagrangian densities which depend on the function and its gradient, namely we try to minimise $\int L(Dw,w,x) dx$. ...
- 469
1
vote
1
answer
3k
views
The conormal derivative of a function
Hi!
I was wondering about the definition of the conormal derivative of a function
$u$ which is given on a domain $\Omega$. It is known that if $-\Delta u = f$, considered
as functionals on $H^1_0(\...
- 13
4
votes
2
answers
904
views
Nash inequality on a compact domain?
I have come across a few papers that make use of the Nash inequality for functions on a compact domain. Unfortunately, nobody cites a reference for the proof of this result. Is going from the ...
- 617
2
votes
1
answer
253
views
A question on Schwartz distributions
I have a question on the tempered distributions, namely, continous functionals on Schwartz class endowed with the weak* topology. Is is a Barreled space, say, a space whose convex, balanced, ...
- 23
4
votes
1
answer
586
views
Upper bounds for the solution of an elliptic PDE depending on a parameter.
Suppose I have the following PDE on $[0,1]^n$
$$\mathcal{L}u = -\nabla \cdot \left(a(x, r)\nabla u\right) = f(x,r), \qquad x\in [0,1]^n,$$
with periodic boundary conditions and $\int f(x) dx =0$ . ...
- 617
2
votes
1
answer
267
views
Fourier transform and spectrum of PDOs in $L^p$
Let $K$ be a compact subset in $\mathbb{R}^n$ with $m(K)=0$, Suppose $supp\hat{u}\subset K$ for some $u\in L^p$,where $2\leq p\leq \frac{2n}{n-1}$,can we get $u\equiv 0$ ?
Motivation: If $K$ is a ...
- 1,644
0
votes
1
answer
488
views
Discrete Sobolev space of $R^n$ valued maps
Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say
$f:\Omega \...
- 17
1
vote
2
answers
1k
views
Existence of solution of a Non-linear PDE via Fixed point theorem
Hi all
I've the following non-linear PDE
$-\Delta Y + Y^3 =U$ on $\Omega \subset R^n $, open, bounded, Lipschitz boundary domain
$Y=0 , $ on $\partial\Omega$
1.Let $Y\in H_0^1 $ and as $H_0^1 \...
- 17
0
votes
1
answer
142
views
A special Integral Kernel
Does there exist either one / general class of non-negative definite , symmetric Integral Kernel map satisfying the following properties ??
$f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$
$K:L^2(\...
- 17
1
vote
1
answer
1k
views
Almost analytic continuation
Let $f\in S^{\alpha}$ for some $\alpha \in \mathbb{R}$(which means that f is smooth and satisfies $|D^{\beta}f|\leq C(1+|x|)^{\alpha-\beta}$),a function $\tilde{f}$ on $\mathbb{C}$ is called an almost ...
- 1,644
4
votes
2
answers
820
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 ...
- 103
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
$$u\in L^2(0,T;W^{1,2}(\Omega))\cap L^\infty((0,T)\times\Omega))\ \text{and } \partial_tu\in L^2(0,T;W^{-1,2}(\...
- 103
3
votes
1
answer
678
views
Is this kernel space of finite dimension ?
Assume that $P \in \Psi^{m}(X)$ (X is a $C^{\infty}$ manifold)is properly supported and has a real principal part p which is homogeneous of degree m.I'm interested in the existence theorem(at least ...
- 1,644
3
votes
1
answer
949
views
ODE continuous dependence on parameters to PDE
I want to learn how to apply certain ODE theory to PDE. If we have a Banach space ODE $$x'(t) = f(t, x(t), p),$$ $$x(0) = x_0$$ where the equation is over same domain $t \in (a,b)$, then via the ...
- 55
2
votes
2
answers
470
views
Linear coupled parabolic PDE system with Holder continuous coefficients
I am interested in proving existence/uniqueness to: find $u(x,t)$, $v(x,t)$ such that
$$u_t - a_1u_{xx} - a_2u_x - a_3u -a_4v = f$$
$$v_t - a_5u_{xx} - a_6u_x - a_7u - a_8v_{xx} - a_9v_x - a_{10}v = g$...
- 55
1
vote
1
answer
2k
views
Basic questions about parabolic Holder space
Hi, I am interested in learning a bit more about this space. I have exhausted all the books available at my disposal, and none of them explain much of the basics for me. Here's a definition of this ...
- 67
1
vote
1
answer
3k
views
How to show this Holder bound?
Define the seminorm on the space $S=[0,1]\times[0,T]$
$$\mid u\mid_{\alpha} = \sup\frac{|u(x, t) - u(y,s)|}{(|x-y|^2 + |t-s|)^{\frac{\alpha}{2}}}.$$
Define the norms on the same space
$$\lVert u \...
- 67
2
votes
1
answer
686
views
Solutions to Heat Equations with Obstacles!
Consider a closed Riemannian manifold $(M,g)$ and a positive function $\psi: M \to R$. Fix a point $p \in M$, I have been struggling to construct a solution to the heat equation, $\partial_t u = \...
37
votes
4
answers
4k
views
Which differential equations allow for a variational formulation?
Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation
$$
\frac{d}{dt}\frac{\...
- 7,583
0
votes
1
answer
612
views
Calculating a distributional derivative
Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to ...
- 213