**1**

vote

**1**answer

163 views

### if $\Pi_1$ and $\Pi_2$ be elliptic planes then $\Pi_1 \oplus \Pi_2 $ is still elliptic?

Let $\Omega \in\Lambda^{4}\big(V^{ \star }\big)$ be volume form. Define symplectic bilinear form
$q: \Pi \oplus \Pi \rightarrow R $
$\big( \alpha ,\beta \big) \longrightarrow \alpha ...

**1**

vote

**1**answer

147 views

### Under which conditions Jacobi PDE system can be represented to symplectic monge Ampere equation?

We know we can reduce Symplectic monge ampere equations to Jacobi PDE system with some compatibility condition. I want to see when the vise versa is correct? and is there any theorem for it.
Here ...

**5**

votes

**1**answer

450 views

### 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 ...

**1**

vote

**1**answer

154 views

### A Cauchy problem for an iterated Euler-Poisson-Darboux eqaution

Good morning,
I'm interested in solving a Cauchy problem for the iterated singular EPD.
Well, Weinstein (On a class of PDEs of even order, 1955) showed how the decomposition formula leads to the ...

**3**

votes

**1**answer

272 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 $\int f(x) dx =0$ . ...

**0**

votes

**1**answer

259 views

### LINEAR Parabolic equations. Smooth dependence from initial data

I am looking for results that show smooth dependence of a solution to a parabolic equation, from the initial data.
More specifically I have the following problem:
CONSIDER spaces $P:=\mathbb{R}^k$ ...

**5**

votes

**1**answer

859 views

### Short time existence on nonlinear parabolic PDE

I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact?
in which book we have this fact, the number of ...

**0**

votes

**0**answers

388 views

### What is the Schauder estimate on usual Hölder space for parabolic type equations

What is the Schauder estimate on usual Holder space for parabolic type equations ?

**3**

votes

**0**answers

114 views

### Quadratic forms over symmetric $3\times3$ matrices

This question is motivated by a technique called compensated compactness (CC), which was elaborated by L. Tartar and F. Murat for the analysis of PDEs. The foreword of CC is a problem about quadratic ...

**2**

votes

**1**answer

146 views

### Differential finiteness of power series with hypergeometric coefficients

Let $f$ be a formal (or convergent near the origin, as you wish) power series with variables $x_1$, ... , $x_n$ and complex coefficients.
$$ f = \sum_I f_I \mathbf x^I $$
This power series is said to ...

**1**

vote

**1**answer

282 views

### Linear inhomogeneous PDE

I am trying to understand the behavior of the following linear PDE:
$$\partial_t u(x,t) = \partial_{xx} u(x,t) + f(x) u(x,t)$$
where I set $f(x)=\lambda e^{-x^2} -1$ and with:
IC : $u(x,0)= 1$ on ...

**8**

votes

**3**answers

3k views

### Physical Interpretation of Robin Boundary Conditions

In a (bounded) domain $\Omega \subset \mathbb{R}^n$, if we're studying the Laplace equation or heat equation or such PDE's we can impose the Dirichlet
$u|_{\partial\Omega} \equiv 0$,
Neumann
$D_{\nu} ...

**4**

votes

**2**answers

205 views

### hodographic transformation

Let $\phi(x,t)$ be smooth function.
Let $\zeta= x-t$ and $\eta= x+t$. $u=\phi_\zeta$, $v= \phi_\eta$.
Let $u$, $v$ satisfies following equations:
1-
$$u_\eta- v_\zeta= 0$$
...

**0**

votes

**0**answers

136 views

### coercivity vs boundedness of operator

hi,
the definition of coercivity and boundedness of a linear operator L between two B spaces looks similar:
$\|Lx\|\ge M _ 1\|x\|$ and $||Lx||\le M _ 2\|x\|$ for some constants $M _ 1$ and $M _ 2$. ...

**1**

vote

**0**answers

173 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 ...

**3**

votes

**1**answer

460 views

### Stochastic Heat Equation

Given the heat equation:
$$\partial_{t}{\varPhi(x,t)}=k^2\partial_{xx}{\varPhi(x,t)}$$
with the boundary conditions:
$$\Phi(x,0)=\Phi_0$$
and a Neumann boundary condition of the kind:
...

**0**

votes

**1**answer

526 views

### Hopf Boundary Point Lemma

The Hopf Boundary Point Lemma
http://en.wikipedia.org/wiki/Hopf_lemma
is a result for the unit normal vector field and the normal derivative.
Is it true if one considers arbitrary directional ...

**7**

votes

**0**answers

312 views

### Regularity of solutions to a linear degenerate parabolic pde

I've encountered the following problem which is causing me some trouble :
Let $(M^2,g)$ be a smooth compact Riemannian surface (with constant curvature for instance) and consider a smooth function ...

**3**

votes

**0**answers

368 views

### Boundary regularity of the solution to the Beltrami equation

Hello, this question might sound a little vague, but I still dare to state , and I am basically requesting for some reference:
Let us consider the orientation-preserving homeomorphic solutions $f: D ...

**1**

vote

**2**answers

493 views

### Heat equation with Neumann BC

Consider the heat equation $u_t=\Delta u$ with Neumann boundary condition in a bounded domain $\Omega$.
Is this true to say:
$$\|u(. , t)-v(. , t)\|_p\leq \|u(. , 0)-v(. , 0)\|_p$$ where $u$ and $v$ ...

**8**

votes

**9**answers

3k views

### Textbooks for PDE between Strauss and Folland

Walter A. Strauss's Partial Differential Equations: An Introduction is a classic PDE textbook for the undergraduate students. While Folland's Introduction to Partial Differential Equations, is a nice ...

**0**

votes

**0**answers

513 views

### characteristic surface

Hello,
I have the following 4 PDEs which I am trying to solve for $G(x,y,z)$ :
(1) $G_{xy}=0$
(2) $G_{xz}=0$
(3) $G_{yz}=0$
(4) $G_{xx}-G_{yy}=0$.
It is not hard to see that the general ...

**2**

votes

**0**answers

259 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 ...

**3**

votes

**1**answer

576 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
$$
u(t,x)= \int_R \frac{e^{-\frac{(x-y)^2}{2t}}}{\sqrt{2\pi t}} \mu(d y)
$$
...

**1**

vote

**3**answers

440 views

### How to prove/disprove that quasiconformal maps send measure-zero sets to measure-zero sets

$Qn#1 $
: Let $f:U\to V$ be a $K$ quasiconformal homeomorphism ( NOT diffeomorphism ) of plane open subsets of $C$. By my definition of quasiconformality, I mean 1)$f$ is continuous, 2)the weak ...

**0**

votes

**2**answers

254 views

### Structure of solutions of a PDE from a game theory problem

I found the following the following differential equation in the context of a Game Theory problem. I was wondering if this is related to any known family of equations or whether there is any hint ...

**2**

votes

**1**answer

377 views

### orthonormal basis of eigenvectors for laplacian on a concave polygon

I am interested in the Laplace operator $\Delta$ on a concave polygon.
When the polygon is convex, it is known that $\Delta: H^2(\Omega) \rightarrow L^2(\Omega)$
is boundedly invertible. In addition, ...

**6**

votes

**3**answers

756 views

### Analytical solution to a Linear advection-reaction PDE

I am looking for an analytical solution for the linear PDE
$(1)\qquad\qquad \qquad f_t+ A f_x + B f = 0, $
Where $A$ and $B$ are constant matrices and $f=f(x,t)$ is a vector.
Clearly each one of ...