All Questions
17 questions
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
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 ...
- 468
1
vote
0
answers
106
views
Solution to hyperbolic linear second order PDE
I am trying to prove the existence (and uniqueness) of a weak solution for a specific PDE. First, let me formulate the problem. I asked the question on the Mathematics page but did not get a solution ...
- 11
2
votes
1
answer
163
views
Justification for uniqueness of solutions to dispersive PDE
For the sake of concreteness, we consider the linear Schrödinger equation
$$
\partial_t u = i\Delta u, \ \ \ \ u(0, x) = u_0(x).
$$
The solution is typically obtained by taking the Fourier transform ...
- 21
1
vote
0
answers
43
views
Mixed boundary condition of parabolic equations
Let $ \Omega $ be a bounded and smooth domain in $ \mathbb{R}^n $. Assume that
$$
\partial\Omega=\partial\Omega_D\cup\partial\Omega_N,
$$
where $ \partial\Omega_D $ and $ \partial\Omega_N $ are ...
- 1,219
2
votes
0
answers
183
views
Regularity of elliptic partial differential equation with mixed Dirichlet-Robin boundary condition, to prove $u\in H^{2}(\Omega)$
I have posted this problem on Math Stackexchange but got no reply.
When I deal with the wave equation with dynamical boundary condition, I am confused by the regularity of the following elliptic ...
- 63.9k
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 $i=1,\ldots, r$, ...
- 108k
0
votes
0
answers
182
views
Has this form of the heat equation been solved for the radiation boundary condition
Below is a solution to a special form of the heat equation. I have found the postings on the heat equation and they are far above my head. I tried to find a tag on Transport Theory for both heat and ...
5
votes
2
answers
273
views
Linear hyperbolic PDE on compact two dimensional domain
Consider the equation
$$
\begin{equation}
\frac{\partial^2f}{\partial x\partial y}=f
\end{equation}
$$
on a Jordan domain (i.e. the interior of a simple, closed curve on the plane). The equation is ...
- 21.5k
5
votes
0
answers
201
views
Where to locate $0\in \Omega$ to get $u_{\varepsilon}(0)\neq 0$ where $\Delta u_{\varepsilon} + (\lambda-\varepsilon) u_{\varepsilon} = \frac{1}{|x|}$
Let $\Omega \subset \mathbb{R}^3$ a smooth bounded domain with $0\in \Omega$ and $u_\varepsilon(x)$ the solution to
$$
\Delta u_\varepsilon + (\lambda-\varepsilon) u_\varepsilon = \frac{1}{|x|}\quad \...
- 2,494
2
votes
1
answer
850
views
The Biharmonic Eigenvalue Problem on a Rectangle with Dirichlet Boundary Conditions
I am interested in solving the following biharmonic eigenvalue problem.
$$\begin{array}{cccc}
& \Delta ^2 \Psi (x,y) = \lambda \Psi (x,y), & - a \le x \le a & - b \le y \le b \\
&...
- 271
3
votes
2
answers
272
views
A Global Estimates for Linear Elliptic PDE
Let $\Omega$ be a bounded smooth region in $R^n$ and $u$ satisfy
$-\Delta u+a(x)u=f, \ \ u|_{\partial \Omega}=0$,
where $a(x)\geq 0$ and $f(x)$ are smooth functions. I wonder if the following ...
- 2,494
2
votes
1
answer
380
views
Best approach to solve this PDE
I need to solve this Partial Differential Equation for $\lambda(x,y)$,
$$\frac{\partial \lambda}{\partial x} + h(x,y)\frac{\partial \lambda}{\partial y} - \lambda \frac{\partial h}{\partial y} = 0$$ ...
- 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 ...
- 3,322
1
vote
0
answers
258
views
Is this Stefan-type problem an open problem?
I am looking for a weak-formulation that would give me an existence and uniqueness of a solution of a Stefan-type problem. It is basically a 2-phase Stefan problem in 2D except that the free-boundary ...
2
votes
0
answers
329
views
General solutions for HJB equations in a special case.
I am reading the book of Wendell Flemming in control theorem to learn the HJB equation
Here is the setting that interests me: Let $g_i: i=1,2$ be $C^2 =C^{2}(-\infty,\infty)$ functions such that $0\...
- 281
3
votes
1
answer
1k
views
Long time behavior of the heat equation on R
Let $\mu\in\mathcal{S}'(R)$ be a Schwartz distribution. The solution of a heat equation with $\mu$ as the initial data is
$$
u(t,x)= \int_R \frac{e^{-\frac{(x-y)^2}{2t}}}{\sqrt{2\pi t}} \mu(d y)
$$
...
- 2,715