All Questions
Tagged with differential-operators linear-pde
17 questions
0
votes
0
answers
87
views
Curl-Div equation with singular matrix
I want to solve the equation:
$$
\begin{cases}
\nabla \times (A \mathbf v)=f, \quad x\in \Omega \\
\operatorname{div}(\mathbf v)=0,
\end{cases}
$$
where $\Omega \subset\mathbb{R}^n$, is an open set, $...
- 617
4
votes
3
answers
473
views
Generalized Fuchsian-type PDE
Consider
$$
\big(1+ t\partial_t\big) \left(\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x\right)A(x,t)+ {t\over (1-x)^3} A(x,t)=0
$$
with the initial condition $A(x,0)=1$. In a small $t$...
- 141
3
votes
1
answer
407
views
Existence of solution to linear inhomogeneous first order PDEs systems
Maybe I am asking a triviality. If that is the case, please let me know and I will close the question. I have searched a lot but I didn't find an adequate and forceful response.
For $i=1,\ldots, r$, ...
2
votes
0
answers
95
views
Laplacian coupled with another equation over a two-dimensional rectangular region
I have the two-dimensional Laplacian $(\nabla^2 T(x,y)=0)$ coupled with another equation which is:
$$\frac{\partial t}{\partial x}+\alpha(t-T)=0 \tag 1$$
where it is known that $t(x=0)=t_i$.
The ...
- 47
4
votes
0
answers
242
views
Equivalent definitions of differential operator
This puzzles me from some time and is in parts connected to the questions Symmetrized derivatives version and Symmetrized derivatives version II.
For me the linear DO between vector bundles $E$ and $...
- 437
4
votes
1
answer
331
views
Existence of solution for the PDE $p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q)$
Consider the following PDE:
\begin{equation}
p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q),\tag{$\star$}
\end{equation}
where $g$ is a flat function at the point (...
- 307
5
votes
1
answer
308
views
The division problem for tempered functions
It is well known (see for example S Łojasiewicz, Sur le problème de la division, Studia Math. 8 (1959), 87–136.) that any linear partial differential operator with constant coefficients is surjective ...
- 193
2
votes
1
answer
127
views
Positive form for a homogeneous elliptic pde
I have a pde of the following form:
\begin{align}
&P(x,D)u = f \text{ on } \Omega, \\
&P(x,D) = \sum\limits_{|\alpha|=2m}a_\alpha(x)D^{\alpha},
\end{align}
where one can assume that $f$ ...
- 537
8
votes
3
answers
858
views
What does the flow of the principal symbol of the differential operator tell us about the PDE?
Disclaimer: Let me apologize in advance for asking this slightly vague question
Let $M$ be a manifold and let $P$ be a partial differential operator acting on $C^{\infty}(M)$. Associated to $P$ there'...
- 7,789
7
votes
1
answer
666
views
A very basic question about projections in formal PDE theory
I am learning formal PDE theory for my research and I am currently struggling to have a basic understanding of the operations involved in completing a (say, linear) PDE system to an involutive one (...
- 7,817
4
votes
0
answers
75
views
The sum of linear partial differential operators of equal strength
If $P$ and $P'$ are linear partial differential operators with constant complex coefficients on $U = \mathring U \subseteq \Bbb R^m$, we say that $P \sim P'$ if and only if $\dfrac {\tilde P} {\tilde {...
- 5,407
3
votes
1
answer
423
views
Harnack Inequality
In the Han and Lin book there is a Harnak inequality for elliptic operators of the kind:
$$
L u = D_i \big(a^{ij}\, D_ju\big),
$$
and the constant $C$ in the Harnack inequality does not depend on the ...
- 823
4
votes
2
answers
481
views
Hörmander's hypoellipticity theorem for complex coefficients
Hörmander's theorem says that if $L = \sum _{i=1} ^r X_i ^2+ X_0 + f$ on some open subset $U \subseteq \Bbb R$ has the property that the Lie algebra generated by $\{X_0, \dots, X_r\}$ at every point ...
- 5,407
1
vote
0
answers
171
views
Existence of solution?
I am sorry if this question is not at the MO level. But I have not found a reference so I would like ask it here.
Follow this paper :http://www.math.ku.dk/~hugger/articles/CTAC2003.pdf
Let $\mathcal{...
- 131
1
vote
0
answers
84
views
Parabolic partial differential equation, initial conditions
Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$.
Given the parabolic PDE $$\partial_tf+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...
- 11
7
votes
2
answers
905
views
Fredholm alternative result for general elliptic system?
Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading ...
- 125
5
votes
1
answer
471
views
Please recommend some literature on the systematical theory of the elliptic systems!
Now I'm interested in the theory of elliptic systems, for example, both the linear and nonlinear case, the exsitence and regularity results, and is there a Fredholm alternative result for the linear ...
- 125