All Questions
Tagged with fa.functional-analysis ap.analysis-of-pdes
1,304 questions
1
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0
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Perturbation in Besov space
$\|f\|_{B^{0}_{p,p}}=(\sum_{j\geq -1} \|\Delta_j f\|_p^p)^{1/p}$ is the Besov norm of $f$.
Here the Fourier transform of $\Delta_jf~(j\geq 0)$ is $\psi(2^{-j}\xi)\hat{f}(\xi)$ and $\psi$ is a smooth ...
- 515
2
votes
1
answer
301
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Simplicity of eigenvalues
Consider the Sturm-Liouville operator$$Au = -(pu')' + qu \text{ on }I = (0, 1),$$where $p \in C([0, 1])$, $p \ge \alpha > 0$ on $I$, and $q \in C([0, 1])$. No further assumptions are made; in ...
- 369
9
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2
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553
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Asymptotic behavior of Sturm-Liouville eigenvalues
I have two questions.
Consider the operator $Av = -v'' + a(x)v$ on $I = (0, L)$, with zero Dirichlet condition and $a \in C([0, L])$.
Let $(\lambda_n)$ denote the sequence of eigenvalues of $A$....
- 369
0
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1
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843
views
$C^{\infty}_{loc}$-convergence - right definition
Let $\Omega \subset \mathbb{R}^{n}$ be some open set. Let $f_{n},f\in C^{\infty}(\Omega)$. My question is: What does the following phrase mean? $f_{n}$ converges to $f$ in $C^{\infty}_{loc}(\Omega)$. ...
- 35
2
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0
answers
110
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Biharmonic Equation in a Rectangle with Some Uncommon Boundary Conditions
I asked this question on Mathematics network but it didn't receive any answers. So I assume it is just beyond the classic things in PDEs and I decided to ask it here too.
Consider the following ...
- 271
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3
answers
320
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Exists $C = C(\epsilon, q)$ such that $\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)}$ for all $W^{1, 1}(0, 1)$? [closed]
Let $1 \le p < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)} \text{ for all }u \in W^{1,...
user81900
1
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0
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83
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Boundedness of a function that satisfies a PDE-type inequality
Let $\Omega$ be a bounded Lipschitz domain, and let $u\colon[-T,0]\times \Omega \to \mathbb{R}$ be a function with $u(-T)=0$.
Suppose that
$$\sup_{-T \leq t \leq 0} \int_\Omega |(u(t)-k)^+|^2 + \int_{...
11
votes
2
answers
712
views
Poincaré lemma for distributions
Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form
$$
u=\sum_{1\le j\le n} u_j dx_j,\quad ...
- 16.2k
1
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0
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177
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How does the $L^\infty$ norm of the solution of $-\Delta u + \lambda u =0$, $\partial_\nu u=\alpha$ depend upon $\alpha$ and $\lambda$?
Let $\lambda > 0$ be a constant and let $u$ be the weak solution on a bounded domain $\Omega$ of
$$-\Delta u + \lambda u = 0 \quad\text{in $\Omega$}$$
$$\partial_\nu u = \alpha \quad \text{on $\...
- 11
1
vote
1
answer
105
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If $u \in L^2(0,T;X_0)$ with $u_t \in L^2(0,T;X_2)$, then is $u \in L^\infty(0,T;X_1)$?
Let $X_0 \subset X_1 \subset X_2$ be continuous embeddings, with $X_0 \subset X_1$ compact.
Suppose $u \in L^2(0,T;X_0)$ with $u_t \in L^2(0,T;X_2)$.
Is then $u \in L^\infty(0,T;X_1)$?
To apply ...
- 11
1
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1
answer
440
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$L^1$ convergence to equilibrium of solutions of heat equation
Let $u$ and $v$ be the weak solutions of
$$u_t - \Delta u = f$$
$$u(0)=u_0$$
and
$$-\Delta v = f$$
$$|\Omega|^{-1}\int_\Omega v =0$$
on a bounded domain $\Omega$, where $u$ and $v$ satisfy homogeneous ...
3
votes
1
answer
2k
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The inverse of Laplacian operator for different orders
I post this question in MSE couple of days before and get no response. So I repost it here for better luck. Thank you!
Let $u,v\in C_c^\infty(\Omega)$ and $\Omega\subset \mathbb R^N$ is open bounded ...
- 679
2
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0
answers
150
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Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $
I have a question that it seems simple but I can not solve it.
Let $\Omega$ be the unit ball centered at zero in $\mathbb{R}^N$, $N>4$. Assume that $C_{0,rad}^{\infty}(\Omega)$ is the space of all ...
- 371
2
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0
answers
234
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Concentration compactness on a compact setting
Consider a compact Riemannian manifold $M$ of dimension $n$ and a sequence of positive functions $\{\varphi_k\}_k \in C^\infty(M)$ such that $\{\varphi_k\}_k$ satisfy the basic concentration ...
- 1,407
2
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0
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491
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$L^\infty-L^1$ smoothing effect for the heat equation
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$.
Let $u \in L^2(0,T;V)$ be the weak solution of the heat equation
$$u_t - \Delta u = 0$$
$$u(0) = u_0$$
where $u_0$ is bounded initial data. Here ...
- 21
10
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1
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973
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$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$
I posted this question first in Math.StackExchange one week ago here, but I didn't get an answer or a helpful comment so I repost it here:
Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded ...
- 261
3
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2
answers
397
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The space $L^p(\partial\Omega)$ in cited references
The space $L^p(\partial\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$ appears in a lot of PDE textbooks without being given any definitions, not even in those with a detailed appendix ...
user14319
1
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0
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351
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Heat semigroup ultracontractive?
Let $g(x,t)= \frac{1}{(4 \pi t)^{\frac{n}{2}}}e^{\frac{-|x|^2}{4t}}$ be the heat kernel on $\mathbb{R}^{n}.$ Is the standard definition now to say that this heat-semigroup $T(t)(f):=g *f(.,t)$ is ...
- 11
6
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1
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357
views
Travelling waves for nonlinear Schrödinger equation
Consider the following nonlinear Schrödinger equation:
$$
-\Delta \Phi - i\frac{\partial \Phi}{\partial t} = f(|\Phi|^2)\Phi,
$$
where $\Delta$ is the Laplacian on $\mathbb{R}^n$, $f$ gives the ...
- 61
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
4
votes
1
answer
328
views
Is Lax-Milgram true without the separability assumption?
I read the Lax-Milgram Theorem in the Navier-Stokes Equations by Temam:
Let $X$ be a separable Hilbert space (norm $\|\cdot\|_X$) and let
$$
a:X\times X\to\Bbb{R}
$$
be a bilinear continuous ...
user14319
0
votes
1
answer
562
views
$H_0^1(\Omega)$ in the study of the Navier-Stokes Equations
This is cross-posted on MSE: https://math.stackexchange.com/q/1584519/9464
Let $\mathcal{V}$ be the space (without topology)
$$\displaystyle \mathcal{V}=\{u\in C_0^\infty(\Omega)\mid \nabla\cdot u=0\...
user14319
4
votes
1
answer
161
views
Commutator representation of certain smoothing operators
I have a question regarding the classical trace $\text{Tr} \colon \Psi^{-\infty}(S^1)\to \mathbb C$ on pseudodifferential operators of infinite negative order (i.e. smoothing operators), defined over ...
- 313
4
votes
1
answer
346
views
Reference for a Heat Process in a Wedge
I would like to ask about an explicit suggestion/reference for the following type of heat processes:
Roughly, assume we have a "wedge" $W$ of the following form - a domain in $\mathbb{R}^n$ with a ...
- 337
1
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0
answers
158
views
On the differentiability of a certain map from $ (0,\infty) $ to $ \Bbb{R} $
This problem arose from my study of energy-conservation for non-linear Schrödinger equations. Suppose that we have the following data:
$ u \in C^{1} \! \left( (0,\infty),{L^{2}}(\Bbb{R}^{n}) \right) $...
- 1,396
5
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1
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481
views
A continuous path between two Sobolev functions
Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Let $u_1$, $u_2\in H^{1}(\Omega)$ such that $T[u_1]=T[u_2]=T[\omega]$ where $T$ stands for the trace operator and $\omega\in H^1(\...
- 679
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0
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470
views
Derivatives of Mollified functions
I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following:
Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...
- 131
1
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0
answers
86
views
Asymptotics of a elliptic pde when exponent gets large
I am interested in the following pde
$$ -\Delta w_p + \left( \frac{1}{p-2} +1 \right) \frac{ | \nabla w_p|^2}{w_p} + \epsilon(p) \left( \frac{1}{w_p} \right)^{(p-2)} = (p-2) w_p $$ in the unit ball $...
- 1,385
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0
answers
244
views
A modification of Minty's trick?
I have the following result:
$$0 \leq \int_0^T (a(t)- |w(t)|)(b(t) - g^{-1}(|w(t)|))\quad\forall w \in L^2(0,T)$$
where $a$ and $b$ are both non-negative.
Does it follow that $b(t) = g^{-1}(a(t))$? ...
- 21
4
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0
answers
500
views
Properties of the solution of the heat equation
Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention.
Note 2: This is my research problem, but the original problem ...
- 679
1
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0
answers
154
views
One-parameter group of unitary operators and Core
Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...
- 11
0
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1
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123
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The monotone operator in $BV$ space
I am considering the following minimizing problem:
$$
\min_{u\in BV(\Omega)}\{\frac12\|u-u_0\|_{L^2}^2 + |u|_{TV(\Omega)}\}
$$
where $u_0\in BV(\Omega)$, $\Omega\subset \mathbb R^2$ is open bounded, ...
- 679
3
votes
0
answers
74
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Trace space of $\{ t^su \in L^2(0,\infty;X) \mid t^su_t \in L^2(0,\infty;Y)\}$ for $s \in (-\frac 12, \frac 12)$
Let $s \in (-\frac 12,\frac 12)$ and let $X=D(\Lambda)$ be a Hilbert space with $\Lambda$ the infinitesimal generator of a bounded semigroup of class $C^0$ in $Y$ (which is another Hilbert space), and ...
- 251
4
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1
answer
180
views
Elliptic regularity for two dimensional domains
Suppose $ \Omega$ is a smooth bounded domain in $ R^2$. I am interested in the regularity of solutions to
$$-\Delta u(x) = f(x) \mbox{ in } \Omega$$ with $ u=0$ on $ \partial \Omega$.
If $ f \in ...
- 1,385
1
vote
1
answer
91
views
degree theory for elliptic equations; special solutions
I am interested in using degree theory to examine some semilinear problems.
But instead of just looking for solutions lets assume i am looking for a certain class of solutions; for instance lets ...
- 1,385
2
votes
0
answers
86
views
Regularity of $u$ in $u_t - \Delta \beta(t,u) = f$, can we get $u_t$ is a function?
I'm looking for reference discussing the regularity of the weak solution $u$ to the equation
$$u_t - \Delta \beta(t, u) = f$$
$$u(0) = u_0$$
where $\beta(t,\cdot)$ is a nonlinear function depending ...
- 51
5
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0
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205
views
Steklov averages in PDE: what to do when we have time-dependent elliptic operator
One may have an equation (with boundary conditions omitted below)
$$u_t - Au = f$$
$$u(0)=u_0$$
which has a weak solution $u \in L^2(0,T;V) \cap C([0,T];H)$ in the sense that
$$-\int_0^T \int_\Omega u(...
- 91
1
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0
answers
205
views
A Question about compactness of an embedding into $L^p$ spaces
Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says
For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have
$$ \Lambda \int_{\Omega} \frac{u^...
- 371
10
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3
answers
1k
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Historical developement of analysis and partial differential equations (especially in the 20th century)
Q: Is there a set of some comprehensive surveys or monographs describing (in
technical detail) the historical development of the various
subareas of analysis and partial differential equations?
I'...
3
votes
0
answers
74
views
Semi-continuity of the dimension of the null space
Suppose $T_n : X \rightarrow X$ is a sequence of Fredholm operators on a Banach space such that $T_k \rightarrow T$ strongly (in the induced operator norm). If $N_k$ and $N$ denote the dimensions of ...
- 537
9
votes
3
answers
2k
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Real analyticity of solution of heat equation
Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...
- 1,407
0
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0
answers
371
views
Harmonic function with Dirichlet boundary condition
Consider the domain $D = \{(x_1, x_2,.., x_n) \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$. Let $D$ be divided into two parts $D_1$ and $D_2$ by the hyperplane $H = \{x_1 = \frac{1}{2}\}$. My question is: ...
- 1
4
votes
3
answers
1k
views
A space of distributions vanishing on the boundary
The revised question
After more reflection on the problem, I might have found the answer by myself. Let $U$ be an open subset of $M$, irrespective of whether it has a boundary or not. Let
$$\mathcal ...
- 5,407
4
votes
1
answer
360
views
Mixed norm estimate for the heat equation
Consider the inhomogeneous linear heat equation
$$\partial_tu-\Delta u=F$$
on $\mathbb R^n\times [0,1]$ (say) with zero initial data. Assume $F$ is very nice (say Schwarz), so that we have a nice ...
- 5,169
5
votes
1
answer
233
views
Zeta-Determinant Theorem
Recently, someone asked on MO about lecture notes from Graeme Segal's "Stanford lectures" on TQFT, and the answer was to check here.
When scrolling over the notes, I stumpled of Prop. 2.8.2 in ...
- 9,853
6
votes
3
answers
481
views
Quantum Mechanics and bilinear optimal control theory
I was wondering whether there are any rigorous results about the optimal controllability of Schrödinger operators.
So my question is something like this:
Let $i \partial_t \psi(x,t) = H_0(x)\psi(x,t)...
- 259
2
votes
0
answers
112
views
Changing frames of the tangent bundle with Schwartz functions [closed]
Let's consider two global frames $\{v_{1},....v_{N}\}$ and $\{u_{1},....u_{N}\}$ of the tangent bundle $T\mathbb{R}^N$.
Now consider the matrix $\{f_{i,j}\}$that change the frame $\{v_k\}$ to $\{u_k\...
- 601
4
votes
2
answers
505
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Eigenfunctions of the Laplacian on singular spaces
Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian $\Delta$...
- 43
12
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0
answers
476
views
Are Sobolev trace spaces equal from both sides of the boundary?
Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure.
Assume $\partial\Omega=\partial\Omega'$.
Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...
- 8,086
6
votes
1
answer
378
views
Compensated compactness for system of conservation laws?
As far as I knew, the method of compensated compactness can be used only for one-dimensional scalar and $2\times 2$ systems of conservation laws, i.e. $u_t+f(u)_x=0$. But if I understood correctly ...
- 105