**1**

vote

**2**answers

420 views

### Existence of solution of a Non-linear PDE via Fixed point theorem

Hi all
I've the following non-linear PDE
$-\Delta Y + Y^3 =U$ on $\Omega \subset R^n $, open, bounded, Lipschitz boundary domain
$Y=0 , $ on $\partial\Omega$
1.Let $Y\in H_0^1 $ and as $H_0^1 ...

**2**

votes

**0**answers

229 views

### Constant in the Poincare inequality for curl square integrable vector fields

$\newcommand{\v}[1]{\boldsymbol{#1}}$For an $u\in H^1(\Omega) = W^{1,2}(\Omega)$ where $\Omega$ is Lipschitz, we have
$$
{\|u - \frac{1}{\Omega} \int_{\Omega} u\|}_{L^2}\leq C
{\|\nabla u \|}
$$
...

**0**

votes

**1**answer

124 views

### A special Integral Kernel

Does there exist either one / general class of non-negative definite , symmetric Integral Kernel map satisfying the following properties ??
$f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$
...

**1**

vote

**1**answer

322 views

### Almost analytic continuation

Let $f\in S^{\alpha}$ for some $\alpha \in \mathbb{R}$(which means that f is smooth and satisfies $|D^{\beta}f|\leq C(1+|x|)^{\alpha-\beta}$),a function $\tilde{f}$ on $\mathbb{C}$ is called an almost ...

**4**

votes

**1**answer

376 views

### Elliptic regularity in $L^1$

Dear all,
I am looking for a good reference for elliptic regularity in $L^1$. To be more precise
Let $\Omega\subset\mathrm{R}^n$ be a bounded smooth domain, let $A$ be a properly elliptic ...

**4**

votes

**1**answer

243 views

### Frobenius theorem with lesser regularity

Clearly, if one is given a $C^1$ sub-bundle $V$ of the tangent space of a smooth manifold $M$, wheather $V$ comes from a $C^2$ foliation of the manifold is decided by the conditions of the Frobenius ...

**3**

votes

**2**answers

153 views

### Decay rate of nonlocal differential operator?

Hi, Moers.
Let $m(\xi) \in S^0$, that is,
$$
|D^\alpha m(\xi)| \leq C<\xi>^{-|\alpha|}, \quad \forall \xi \in R^n.
$$
It's well known that $m(D)$ is bounded in $L^p$ for $1 < p < \infty$.
...

**3**

votes

**1**answer

546 views

### Alexandrov-Bakelmann-Pucci maximum principle

The ABP maximum principle states (roughly) that, if $a^{ij} \partial _i \partial _j u \geq f$, over a domain $\Omega$ in $\mathbb{R}^n$ (where $a^{ij} \geq C Id >0$), then (assuming sufficient ...

**0**

votes

**0**answers

209 views

### How can we formulate maximal time $T$ in Hyperbolic Kahler Ricci ﬂow

In general, the exact maximal time $T$ of a Riemannian Ricci flow may not be easy to find. However, fortunately, for Kähler-Ricci flows, the maximal time of existence $T$ is explicitly determined by ...

**1**

vote

**2**answers

434 views

### The integrability of fundamental solution of laplace equation follows from integrability of f ?

Hi, I am really struggling with this question.
The question is :
Let $f:R^3\to R$ and $f\in L^2(R^3)$. $f$ is supported on a ball of radius 1/2 centred at origin. Let $u$ be the solution to $\Delta ...

**2**

votes

**0**answers

323 views

### How to apply Lagrange Multipliers to BCs of Time Dependent problems using finite elements?

I am trying to implement a finite element scheme using the method of lines (finite difference in time and finite element in space) and enforcing boundary conditions using Lagrange Multipliers. This ...

**3**

votes

**2**answers

539 views

### Elliptic regularity on bad domain

Let $\Omega \subset R^n$ be an open bounded set. Consider the Dirichlet problem
$$
-\triangle u = 0, x \in \Omega, \quad u = \varphi, x \in \partial \Omega.
$$
If $\varphi$ is a continuous function, ...

**5**

votes

**2**answers

440 views

### $W^{2,p}$ or $W^{1,q}$ regularity for the laplace on a euclidean sphere

Hi,
it is easy to prove the $W^{2,2}(\mathcal S^2)$ regularity for the laplace on the (2 dimensional-) standard sphere $\mathcal S^2:=\lbrace x \in\mathbb R^3: \vert x\vert=1 ...

**5**

votes

**3**answers

457 views

### Continuity with values in L^2

Hi,
let $T>0$, $\Omega\subset\mathrm{R}^n$ be a bounded smooth domain and suppose
$$u\in L^2(0,T;W^{1,2}(\Omega))\cap L^\infty((0,T)\times\Omega))\ \text{and } \partial_tu\in ...

**1**

vote

**0**answers

130 views

### Compactness of solutions of elliptic equation

Consider the following nonlinear elliptic equation
$$
-\triangle u + u + u^3 = g, \quad x \in R^3.
$$
If $g \in L^2(R^3)$, then the set $Q$ of solutions of above equation is bounded in $H^2(R^3)$, and ...

**3**

votes

**1**answer

387 views

### Is this kernel space of finite dimension ?

Assume that $P \in \Psi^{m}(X)$ (X is a $C^{\infty}$ manifold)is properly supported and has a real principal part p which is homogeneous of degree m.I'm interested in the existence theorem(at least ...

**6**

votes

**2**answers

420 views

### Implications of a hypothetical blow-up of Navier-Stokes for the mathematical model

Let us suppose that there exists a (initially smooth) solution of NSE that blows up in finite time. Then, in particular, the corresponding velocity field becomes unbounded as time progresses. Which ...

**1**

vote

**1**answer

186 views

### Reference to the Existence and Uniqueness of the PDE system

Hi all
I've the following Problem on systems of Partial Differential Equations.I have " N " Physical variables. and Finally I form the equation on a bounded domain having regular boundary in R^d. ...

**0**

votes

**3**answers

455 views

### Analyticity of the solutions of PDE

Let's consider a (partial) differential equation with analytic coefficients. The initial conditions may be non-analytic*.
Is it possible a solution $f$, which is non-analytic at any point of the ...

**2**

votes

**1**answer

311 views

### ODE continuous dependence on parameters to PDE

I want to learn how to apply certain ODE theory to PDE. If we have a Banach space ODE $$x'(t) = f(t, x(t), p),$$ $$x(0) = x_0$$ where the equation is over same domain $t \in (a,b)$, then via the ...

**3**

votes

**1**answer

334 views

### What do we know about the semigroup $e^{it\sqrt{-\Delta}}$

I'm very interested in the properties of the semigroup $e^{it\sqrt{-\Delta}}$, it may has some fundamental differences(such as the kernel) with the well-known schrodinger semigroup $e^{it\Delta}$.
...

**2**

votes

**2**answers

296 views

### Linear coupled parabolic PDE system with Holder continuous coefficients

I am interested in proving existence/uniqueness to: find $u(x,t)$, $v(x,t)$ such that
$$u_t - a_1u_{xx} - a_2u_x - a_3u -a_4v = f$$
$$v_t - a_5u_{xx} - a_6u_x - a_7u - a_8v_{xx} - a_9v_x - a_{10}v = ...

**1**

vote

**2**answers

270 views

### how to solve a singular integral equation involving the kernel $1/x$

Dear all,
Suppose we know that $f(x)$ is nonnegative and Hölder continuous at zero with exponents $1/2$. We also know that
$$
f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0,
$$
...

**3**

votes

**1**answer

360 views

### Does the operator $\mathrm{id}-t\Delta$ or its Green's function have a name?

Consider a Riemannian manifold and let
$\mathrm{id}$ be the identity operator, let
$\Delta$ be the scalar, negative-semidefinite Laplace-Beltrami operator, and let
$t > 0$ be a parameter.
Does ...

**1**

vote

**0**answers

199 views

### Invariance of a tensor Laplacian

Let $\phi : \Omega \to \Omega'$ be an invertible mapping between two bounded domains in $\mathbb{R}^{n}$ (typically with $n=2$ or $n=3$), and let $F$ be its derivative (i.e. the Jacobian matrix). Let ...

**4**

votes

**1**answer

162 views

### Integrability conditions for 'componentwise' systems of linear PDEs

I find myself staring blankly at a system of PDEs in $n$ dimensions which has "one equation per component" of the Hessian of the unknown function - that is, it specifies the Hessian in terms of the ...

**1**

vote

**1**answer

367 views

### Basic questions about parabolic Holder space

Hi, I am interested in learning a bit more about this space. I have exhausted all the books available at my disposal, and none of them explain much of the basics for me. Here's a definition of this ...

**3**

votes

**3**answers

599 views

### Classification of a certain System of Linear First Order PDEs whose characteristic polynomial has one real and two complex conjugate zeros

In my research work, I recently have come across the following system of three linear first order pde's whose characteristic polynomial has one real and two complex conjugate zeros.
...

**1**

vote

**1**answer

518 views

### How to show this Holder bound?

Define the seminorm on the space $S=[0,1]\times[0,T]$
$$\mid u\mid_{\alpha} = \sup\frac{|u(x, t) - u(y,s)|}{(|x-y|^2 + |t-s|)^{\frac{\alpha}{2}}}.$$
Define the norms on the same space
$$\lVert u ...

**0**

votes

**1**answer

498 views

### Wave equation v.s.Schrödinger equation

The motivation of comparison of this two kind of operators is that,$$\partial_{tt}-\Delta=(\partial_{t}-i\sqrt{-\Delta})(\partial_{t}+i\sqrt{-\Delta})$$
From the above that a wave operator can be ...

**1**

vote

**1**answer

244 views

### Seeking reference on regularity theory for nonlinear elliptic PDE

Hello,
I am searching for a reference on a result I know must exist proving regularity for weak solutions of a (nonlinear, but well-behaved) elliptic homogeneous PDE. Working over say a bounded ...

**2**

votes

**0**answers

215 views

### Recovering full regularity by energy method in the heat equation

Consider the heat equation
$$
u_t = u_{xx} + f,
$$
on the circle, and for a finite time interval. From Duhamel's principle one can deduce that $u\in L^\infty H^2$ if for instance $f\in L^\infty H^s$ ...

**0**

votes

**0**answers

149 views

### A-priori bound on parabolic PDE that doesn't depend on end time

I have a PDE
$$u_t = a(x,t)u_{xx} + b(x,t)u_{x} + c(x,t)u + f$$
where the coefficients are in parabolic Holder space $\widetilde{C}^{0, \alpha}(I \times [0,T])$ where $I=[0,2\pi]$. The a-priori bound ...

**0**

votes

**2**answers

275 views

### Fundamental Solutions with compact support (distributions)

Assume that we have a differential operator such as $-\frac{\partial}{\partial x^2} + id$ on $\mathbb{R}^1$
We also then argue that if a fundamental solution has compact support, then it is supported ...

**0**

votes

**0**answers

121 views

### Coupled system of linear parabolic PDEs

Hi,
Are there any existence results for the coupled system of linear parabolic PDEs:
$$u_t - a_1u_{xx} - a_2u_x - a_3u = f_1$$
$$v_t - a_3u_{xx} - a_4u_x - a_5u - a_6v_{xx} - a_7v_x - a_8v = f_2$$
...

**1**

vote

**1**answer

220 views

### Heat equation of spatial complex variable

Suppose that $v(t, z)$ is analytic with respect to the complex variable $z$ and differentiable with respect to the real variable $t$ and satisfies the partial differential equation
$$\frac{\partial ...

**0**

votes

**1**answer

273 views

### W^{2,1} REGULARITY FOR SOLUTIONS of Monge-Ampere equation

In the paper by GUIDO DE PHILIPPIS AND ALESSIO FIGALLI:
http://arxiv.org/abs/1111.7207
They proved the $W^{2,1}$ estimate for standard monge-ampere equation
$detD^{2}u=f$
with $f$ bounded from below ...

**1**

vote

**1**answer

312 views

### Generalized Friedrichs Lemma

Taylor's PUP book on pseudodifferential operators in II.7 has an extension of the pseudodifferential version of Friedrichs' lemma to generalized Friedrichs' mollifiers $J_\epsilon$ on a compact ...

**0**

votes

**1**answer

116 views

### Reference: DaPrato and Grisvard parabolic PDEs.

Has anyone read G. DaPrato and P. Grisvard Equations d'evolution abstraites nonlineaires de type parabolique?
It's not available in my library. I am wondering if it's worth me acquiring it: is it ...

**0**

votes

**1**answer

270 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 $P:=\mathbb{R}^k$ ...

**1**

vote

**1**answer

477 views

### Solutions to Heat Equations with Obstacles!

Consider a closed Riemannian manifold $(M,g)$ and a positive function $\psi: M \to R$. Fix a point $p \in M$, I have been struggling to construct a solution to the heat equation, $\partial_t u = ...

**0**

votes

**1**answer

230 views

### Map with prescribed Jacobian

Recently I came up with the following problem.
Suppose $U$ is an open subset of $\mathbb{R}^n$ and we are given a continuous map $M:U\to GL(n;\mathbb{R})$. Does anybody know if there are conditions ...

**1**

vote

**1**answer

423 views

### Tensor analysis/Differential forms outside physics

There are many "geometric systems" like tensor analysis or differential forms calculus, which more or less different perspectives onto the same abstract relations.
Most applications are physical, ...

**2**

votes

**1**answer

176 views

### Continuation of hyperbolic Laplacian eigenfunction

The following question arises while I'm reading a paper of Jerzy Kaczorowski and Alberto Perelli (A correspondence theorem for L-functions and partial differential operators, Publ. Math. Debrecen ...

**-1**

votes

**1**answer

176 views

### Nearly elliptic equations [closed]

If you have a second order elliptic equation but the coefficients of the second order terms only form a nonnegative (instead of positive definite) matrix, then, do you know if there is any literature ...

**1**

vote

**0**answers

299 views

### Relation between interpolation spaces and besov spaces

Consider the following two norms:
The interpolation norm:
1) $\|u; [L_2,\dot H_1^{\infty}]_{1/3,\infty}\| := \sup_{s > 0} \inf_{u=u_0+u_1} \frac{\|u_0\|_{L^2}}{s^{1/2}} + s \|\partial_x ...

**4**

votes

**1**answer

220 views

### Minimizing action squared versus action

I have a very basic question in the calculus of variations:
Suppose I want to minimize the functional
$$A[r, r'] = \int_\Omega L(r, r') dx $$
When is it possible to say that extremals of $A$ agree ...

**1**

vote

**0**answers

61 views

### When can a perturbation be treated as a regular perturbation?

I am working with cauchy problem of the form
$$ ( - \partial_t + A^\delta) u^\delta = 0 , \qquad u^\delta(0,x) = h(x), $$
where the domain of $u^\delta$ is $[0,\infty) \times \mathbb{R}$. The ...

**2**

votes

**3**answers

696 views

### Question About Harmonic Function Theory

Given a non-negative function $u $ defined on $\mathbb{R}^2 $ , and satisfies :
$ \Delta u \leq 0 $ .
How can I prove that $u$ must be constant?
Is there an easy way to do it ?
Thanks !

**24**

votes

**3**answers

1k views

### Which differential equations allow for a variational formulation?

Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation
$$
...