Questions tagged [ap.analysis-of-pdes]
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
4,275
questions
7
votes
1
answer
881
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Lax pairs for linear PDEs
I'm trying to understand the discussion around equation (2.1) of the paper http://www.jstor.org/stable/53053. It says that the linear PDE $M(\partial_x,\partial_y)q=0$ with constant coefficients has ...
- 253
3
votes
1
answer
715
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
2
votes
0
answers
101
views
Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform II
This is a modification of a previous question.
Question: Suppose $a(s)\in C^\infty([0,1])$ and $H(s,x)\in C^\infty([0,1]\times [0,1])$ with $H(s,x)>0$, $\forall s,x\in [0,1]$. Suppose,
$$\sup_{\...
- 458
11
votes
3
answers
1k
views
Boundedness of the derivative of the trace of an H^1 function
As a research preface, this question is linked to a problem of increasing magnetism in Ginzburg-Landau equations that I have distilled for the purpose of getting to the bottom of this technical matter....
- 862
1
vote
0
answers
204
views
Looking for higher order Sobolev inequality
Hello,
On a compact (without boundary) Riemannian manifold (eg. some surface in $\mathbb{R}^n$), I'm looking for a result like
$$\lVert \nabla u\rVert_{L^2}^2 \leq \epsilon\lVert u\rVert_{H^1}^2+\...
- 29
1
vote
1
answer
341
views
A 'conjecture' on critical elliptic pde
I conjecture the following.
Let $\Omega=\mathbb{R}^3-\overline{B_1(0)}$. Define
$$E_{\Omega}(u)=\frac{1}{2}\int_{\Omega}|\nabla u|^2dx-\frac{1}{6}\int_{\Omega}|u|^6dx.$$
$E_{\mathbb{R}^3}$ is defined ...
- 43
15
votes
1
answer
844
views
Symbols of elliptic operators
First let me state the problem, then I'll explain its origin and finally, I'll ask the main question..
Problem S. Fix a positive integer $n$. Find all the pairs $(V, S)$, whith the following ...
- 34.1k
1
vote
0
answers
126
views
A critical elliptic PDE
I am considering the problem $-\Delta u=|u|^4u$, $x\in \Omega\subset \mathbb{R}^3$, $u|_{\partial \Omega}=0$. Where $\Omega$ is a unbounded domain. Some special case like $\Omega=\mathbb{R}^3-B_1(0)$, ...
- 43
2
votes
2
answers
509
views
The logarithmic fast diffusion equation in one space variable with periodic boundary conditions.
I need to know about this non-linear logarithmic fast diffusion equation for a function $u(x,t)$ of one space variable $x$ and time $t$:
$$ u_t = (\ln u)_{xx}$$
which is to run on an interval $ a \leq ...
- 23
2
votes
0
answers
259
views
Best constant of Gagliardo-Nirenberg inequality in exterier domain
In $\mathbb{R}^N$, we know that the best constant of Gagliardo-Nirenberg is characterized by the solution $Q$ of $-\Delta u+u=|u|^2u$ with minimal mass. One have
$||u||_4^4\leq C||u||_2||\nabla u||_2^...
- 43
3
votes
1
answer
307
views
The distribution of roots of elliptic polynomial
If $p(x)$ is an $n$ variables polynomial of even degree with complex coefficients which satisfies the strong elliptic condition, that is, Re$p(x) \ge C|x|^{2m}$ for any $x \in \mathbb R^n$ where $2m$ ...
- 1,361
1
vote
1
answer
473
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
1
vote
2
answers
320
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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 $S$ ...
- 45
4
votes
1
answer
255
views
variation of the obstacle in the obstacle problem
Suppose $D \subset \Bbb C$ with smooth boundary. Let $f \in C^{1,1}(D)$. Let $\varphi$ be the supremum of all members in the set
$$\lbrace g \in C^{\infty}(\overline{D})| g \ is \ subharmonic \ and \...
- 1,171
2
votes
2
answers
392
views
how many nonparabolic ends guarantee a nonconstant harmonic function on Riemannian manifold?
M is a Riemannian manifold.An end E is said to be a non-parabolic end if it admits a positive Green's function with Neumann boundary conditon on E.Otherwise,it is a parabolic end.If M has more than ...
- 689
1
vote
1
answer
1k
views
weak derivative and continuous function
Let $\Omega \subset \mathbb{R}^n$ be a compact smooth hypersurface. Suppose $\varphi \in C_c^\infty(0,T; H^1(\Omega))$ is a $H^1(\Omega)$-valued test function (so $\varphi(t) \in H^1(\Omega)$ for each ...
- 107
11
votes
3
answers
2k
views
Space of sections of a fibre bundle with non-compact base space
Let $\pi: E \rightarrow M$ be a fiber bundle over the manifold M and denote by $\Gamma(E)$ the space of smooth sections of $E$.
For compact $M$ it is well known (Hamilton 1982, Part II Corollary 1.3....
- 5,472
1
vote
2
answers
765
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Fourier transform of function on compact set and Sobolev norm equivalence
Hi all. My question on M.SE is unanswered (https://math.stackexchange.com/questions/254970/fourier-transform-of-function-defined-on-subset-of-mathbbrn) so I want to post it here. I changed it slightly....
- 113
8
votes
3
answers
847
views
abstract evolution equations
Hi
Whenever I read a book on evolution equations, they set up, say the parabolic PDE
$$\dot{y} = Ay + f$$
in abstract function spaces (eg. $L^2(0,T;V)$ and $L^2(0,T;V^*)$). In examples, they always ...
- 107
6
votes
2
answers
164
views
Conditions on a unit vector field to be the Gauss map of some surface immersed in R^3?
Let $U$ be a bounded domain in $R^2$ and let $n : U \to S^2$. Which (necessary/sufficient) conditions must $n$ satisfy in order that there exist an immersion $f : U \to R^3$ such that $n(x)$ is the ...
- 63
2
votes
1
answer
890
views
The PDE $u_t=u_{xx}-u_{yy}$: The simplest linear second-order PDE that isn't elliptic, parabolic, or hyperbolic.
I know that there have been several questions on here and stackexchange about linear PDE's which don't fall into the standard classification, but I had a more focused question which I haven't seen ...
3
votes
1
answer
356
views
Exponential decay for the gradient of a solution
Dear all,
I would like to prove the exponential decay of the derivatives of a solution to the following equation in $\mathbb{R}^N$:
$$
\sqrt{-\Delta+m^2} u +u= f(u),
$$
where I can assume that $m \neq ...
- 31
29
votes
1
answer
3k
views
The Riemann zeros and the heat equation
The Riemann xi function $\Xi(x)$ is defined, with $s=1/2+ix$, as
$$
\Xi(x)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)=2\int_0^\infty \Phi(u)\cos(ux) \, du,
$$
where $\Phi(u)$ is defined as
$$
2\sum_{...
- 10.8k
9
votes
2
answers
1k
views
PDE with the Jacobian Determinant
Hello,
Could you please help me in answering the following question?
Initially I thought that the following problem can be solved through Monge-Ampere equation, but with Monge-Ampere, I have not ...
- 895
5
votes
1
answer
631
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
2
votes
1
answer
340
views
Please recommend some classical books or articles on the compressible Euler equations !
Please recommend some classical books or articles on the local well-posedness result of compressible Euler equations ! The main aim is that I want to learn some basic methods and techniques about the ...
- 35
5
votes
1
answer
392
views
C^\infty versus semiclassical wavefront sets
Zworski states that if $u$ is a compactly supported distribution, independent of the semiclassical parameter $h$, then the relationship between the $C^\infty$ and semiclassical wavefront sets of $u$ ...
- 115
6
votes
0
answers
141
views
Lagrangean uniqueness versus Eulerian uniqueness
(1) Lagrangean description. Let us consider a $N\times N$ system of autonomous ODE:
$$
\dot x=a(x),\quad \mathbb R\ni t\mapsto x(t)\in \mathbb R^N,\quad a:\mathbb R^N\rightarrow \mathbb R^N.
$$
...
- 15.2k
0
votes
0
answers
198
views
damped wave equation
For $t>0$, $x$ in a compact Riemannian manifold $(M,g)$, and $a\in C^\infty(M)$, $a\geq0$, $(\partial_t^2+a\partial_t-\Delta_g)u=0$ is called the damped wave equation.
My question is...why is the ...
- 115
2
votes
1
answer
248
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Distributional limits concerning the regularity of Maxwells equations
This question is related to my previous question about the regularity of the Maxwell equations.
Assume we are working on a space where there are only electric point charges, $(q_i)$, and a blob of ...
- 455
125
votes
15
answers
15k
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Does Physics need non-analytic smooth functions?
Observing the behaviour of a few physicists "in nature", I had the impression that among the mathematical tools they use a lot (along with possibly much more sofisticated maths, of course), ...
- 22.7k
6
votes
3
answers
818
views
Divergence form Elliptic PDE Removable Singularity/Regularity Question
Idea
Given a $W^{1,2}$ solution to a linear divergence form uniformly elliptic pde with bounded coefficients, standard De Giorgi-Nash-Moser theory tells us that the solution is infact (Holder) ...
- 1,771
6
votes
1
answer
1k
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propagation of singularities & the Schrodinger equation
I've been thinking about the following propagation of singularities result:
Let $X$ be a compact manifold, and let $P$ be a differential operator (of, say, order $m$) on $X$ whose principal symbol $\...
- 115
6
votes
1
answer
2k
views
fixed point arguments in PDE
I was curious whether anyone knows of some examples in PDE where
a standard fixed point argument fails to show the existence of a solution but one can apply
one of the more advanced fixed point ...
- 61
1
vote
1
answer
259
views
Does this PDE has a general solution? [closed]
$$K\frac{\partial }{{\partial x}}(h\frac{{\partial h}}{{\partial x}}) = \mu \frac{{\partial h}}{{\partial t}}$$
K and u are constants.
If no,how to get a asymptotic solution?ie,linearize
- 689
0
votes
1
answer
985
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
7
votes
4
answers
630
views
Is there a good metric under which a sequence of compact sets can converge to an infinite dimensional set?
I have a sequence of finite-dimensional, compact sets in $L^2(\Omega)$, where $\Omega\subset \mathbf{R}^2$ is closed and bounded. The dimension grows monotonically with the sequence, and there is no ...
- 71
1
vote
2
answers
849
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 ...
- 1,451
14
votes
1
answer
2k
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Regularity of the Maxwell equations
As is well-known, the Maxwell equations can be phrased vectorially as,
\begin{align}
\nabla \cdot \mathbf E &= \frac{\rho_f}{\varepsilon}, &\text{Gauss's law,}\\\
\nabla \cdot \mathbf ...
- 455
4
votes
1
answer
651
views
Solution of Helmholtz-Equation where Phase is restricted by additional PDE
Hello!
Let's say I have
$(\partial_x^2 + \partial_y^2 + a)f(x,y)=0$
with $f(x,y) \in \mathbb{C}$, ($\lim_{x,y \to \infty} f(x,y)=0$).
Now separate the Amplitude and Phase of the solution:
$$f(x,y)=...
- 41
19
votes
2
answers
2k
views
The speed of gravitational waves in general relativity
Is it possible to mathematically prove that the speed of gravitational waves in general relativity equals the speed of light, without linearizing the Einstein field equations? The approach via the ...
- 816
1
vote
1
answer
79
views
Finding singular "solutions" to the Dirichlet problem for Schrödinger operators that do not admit smooth solutions
Suppose that $\Omega\subset \mathbb{R}^n$ is a smooth open region and that $V:\Omega\to \mathbb{R}^+$ be a positive smooth function.
Then we have a family of operators
$$L_\epsilon =-\Delta -\epsilon ...
- 2,279
3
votes
1
answer
756
views
Change in neumann boundary conditions through coordinate transformation of elliptic PDE, weak formulation
The standard weak formulation of the Neumann problem for the Poisson equation
is to find $u \in H^1 ( \Omega)$ such that for every $v \in H^1 ( \Omega)$:
$$ \int_{\Omega} \nabla u \nabla v d x = \...
- 225
1
vote
0
answers
154
views
two polynomial equations
Let $f:\mathbb R^2\rightarrow\mathbb R$ be a smooth function such that for every point $(x,y)\in\mathbb R^2$ the system
$$f_{11}+2tf_{12}+t^2f_{22}=0$$
$$f_{111}+3tf_{112}+3t^2f_{122}+t^3f_{222}=0$$
...
- 1,359
3
votes
1
answer
252
views
Subharmonic envelope
I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me know....
- 1,171
7
votes
1
answer
518
views
Green functions on Riemann surfaces
Let $(M,g)$ be a compact Rieamnnian surface without boundary and $\Delta_g$ be the Lapalce operator. We note $\lambda_i$ and $\phi_i$ the eigenvalues and eigenunctions of $\Delta_g$. Let also $G_g$ ...
- 71
1
vote
1
answer
182
views
Well-posedness of Euler-Poisson system for semiconductors
Is there anyone can recommend me some important literature references about the well-posedness of both Cauchy problem and initial boundary value problem of Euler-Poisson system for semiconductors? ...
- 35
0
votes
0
answers
293
views
hitting probability for integrated Ornstein-Uhlenbeck process
Consider an Ornstein-Uhlenbeck position process:
$dV_t=dB_t-\lambda V_tdt$
$dX_t=V_tdt$
where $B_t,V_t,X_t$ are all in $R^d$ with $d\geq 3$. Let $X_0\neq0$, $V_0=0$ .
Let $r>0$ and $S_r$ be the ...
2
votes
2
answers
631
views
Easy question on Sobolev spaces
I understand that this question would be trivial for experts, sorry for that, I just need to clarify things.
So let $S(\mathbb{R}^n)$ denote the Schwartz space on $\mathbb{R}^n$ and $W_p$, $W_q$ are ...
- 1,297
2
votes
1
answer
853
views
Weak divergence implies weak differentiability of components?
Suppose $\Omega$ is an open set in $\Bbb{R}^N$ and $\sigma : \Omega \to \Bbb{R}^N$ is a field with all components belonging to $L^2(\Omega)$.
We say that $\sigma$ has weak divergence if there exists ...
- 2,102