**2**

votes

**1**answer

248 views

### Best approach to solve this PDE

I need to solve this Partial Differential Equation for $\lambda(x,y)$,
$$\frac{\partial \lambda}{\partial x} + h(x,y)\frac{\partial \lambda}{\partial y} - \lambda \frac{\partial h}{\partial y} = 0$$ ...

**7**

votes

**2**answers

258 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 ...

**4**

votes

**1**answer

168 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 ...

**2**

votes

**1**answer

267 views

### Uniqueness of classical solution with degenerate boundary

Consider heat equation on the domain $\Omega = (0,1)\times (0,1)$
in the form of
$$ \partial_{t} u = \frac 1 2 x^{3} (1-x) \partial_{xx} u, \quad
(x,t) \in \Omega$$
with initial data
$u(x,0) = x$ for ...

**0**

votes

**1**answer

100 views

### The sufficient condition of the Neumann problem

How can we prove that if $\int_U{f}=0$,then the homogeneous Neumann problem $\Delta u=f$on U,and $\frac{\partial u}{\partial n}=0$ on $\partial U$ has a weak solution in $H^1(U)$?

**0**

votes

**1**answer

166 views

### Concerning Fritz John's article, The Ultrahyperbolic Differential Equation With Four Independent Variables

I am trying to read Fritz John's article, here:
http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.dmj/1077490637
And for the proof of Thm 1.1 in page ...

**2**

votes

**1**answer

131 views

### The maximum in the Poisson problem on the cube with constant source

Question:
Let us consider the Poisson problem on the square with constant source $1$
$$
\begin{cases}
- \Delta u &= 1, \qquad \text{ in } (0,1)^n \\\\
u &= 0, \qquad \text{ on } \partial ...

**1**

vote

**1**answer

305 views

### Solving Stokes Equations using 3D Fourier transforms

How do you calculate the inverse Fourier transform of $\frac{k_ik_j}{k^4}$. I know it has to be a matrix of the form $=δ_{ij}A(r)+r_ir_jB(r)$, but how do you calculate the functions A(r) and B(r)?
I ...

**3**

votes

**0**answers

228 views

### well-posedness of the transport equation

I asked this question before on math exchange but did not have any luck with an answer. I would like to consider a simple example but get a thorough understanding of the theory behind it. I am ...

**6**

votes

**2**answers

263 views

### Uniform bound on the eigenfunctions of the Laplacian

Is it possibly to have $L_\infty$ bounds on the eigenfunctions of the Laplacian operator on bounded regular domains with Dirichlet condition? I found several papers by Sogge but these are pretty ...

**6**

votes

**0**answers

327 views

### 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 ...

**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 ...

**7**

votes

**1**answer

548 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

157 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

278 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

265 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

868 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 ...

**3**

votes

**0**answers

117 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

290 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

207 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

137 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

174 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

476 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

538 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

323 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

499 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

529 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

261 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

580 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

444 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

255 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

380 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

769 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 ...