# Tagged Questions

**2**

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149 views

### A limit involving a regularizing kernel

I am studying the following article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/#
...

**2**

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**0**answers

84 views

### Asymptotics of quasilinear elliptic equations with Dirac right hand side

On a small open neighborhood $U$ of $0 \in \mathbb{R}^n$, consider the quasilinear (possibly monotone) elliptic, scalar PDE in divergence form
$$\nabla \cdot(a(x,u,\nabla u)\nabla u) = \alpha ...

**2**

votes

**0**answers

159 views

### Integrability of ground state solution for elliptic equation

For the solution of semi-linear elliptic equation, for example I'm considering the 2D cubic nonlinear Schroedinger equation, the correspongding elliptic equation is $\Delta u+u^3=u$, with $u>0$. By ...

**2**

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**0**answers

303 views

### Variational Formulation of Boundary Value Problems With Unknown on the boundary

Suppose that we have a linear operator equation on $\Omega$ with Lipschitz boundary $\partial \Omega$,
\begin{eqnarray}
Lu &=& \frac{\partial u}{\partial t},
u(x,0) &=& u_0 \; \; ...

**2**

votes

**0**answers

186 views

### Core of divergence form operator with unbounded coefficient

Consider the unbounded operator $L$ on $L^2(\mathbb{R^d})$ to be the self-adjoint extension of
$$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$ on $C^2_c(\mathbb{R^d})$.
I also assume that ...

**2**

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**0**answers

242 views

### Equation of motion of a charged string in a twisted torus.

The flat torus background:
Say we want to study the sigma model of strings (closed strings $S^1= \mathbb{R}/\mathbb{Z}$) on a flat torus (for example $T^3=\mathbb{R}^3/\mathbb{Z}^3$ with a flat ...

**2**

votes

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542 views

### Comparision of cubic hermite finite element and cubic B-spline finite element (in condition nunmbers of stiffness matrix, or sth else)

Background
Consider the one dimensional second order elliptic PDE,
$$
\left\{\!\!
\begin{aligned}
& -(a(x)u'(x))'+b(x)u(x)=f(x)\qquad x\in[0,1]\\
& u(0)=u(1)=0
\end{aligned}
...

**2**

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148 views

### Regularity properties of the derivatives of a particular function on $D \times D\to \bar{D} $ ?

This question might sound a little less rigorously formulated, but I hope the question still makes sense.
Let $h: S^1 \to S^1$ be an oriention-preserving homeomorphism and let $p(z,t) = ...

**2**

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**0**answers

291 views

### A free boundary problem by finite difference method

I wanna discretize the following free boundary problem
Find $u$ and $\Omega$ such that $\Delta u=1-\delta_0$ in $\Omega$ with the conditions $u=|\nabla u|=0$ on $\partial \Omega$.
I apply finite ...

**2**

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**0**answers

272 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 ...

**2**

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**0**answers

240 views

### Can the solution manifold for an exterior differential system be represented using alternating multivectors?

Differential equations can be written as an ideal of n-forms. Solutions are manifolds where the forms pull back to zero. Is it possible, or useful, to represent the solution by multivectors? For ...

**2**

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**0**answers

302 views

### Vanishing solution of the Poisson equation at infinity

Hi, I am interested in finding some vanish bahavior at infinity of the solutions of this kind of equations:
$-\Delta\phi+a(x)\phi=b(x)$
where $a(x), b(x)\in L^{p}$ with $1\leq p\leq 3$. Besides ...

**2**

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**0**answers

290 views

### Poisson problem with a “scaled” Laplacian.

Let $d_1$ and $d_2$ be positive constants. I'm considering a 2D Poisson-like problem of the form
$$ d_1\frac{\partial^2 u}{\partial x_1^2} + d_2\frac{\partial^2 u}{\partial x_2^2} = f$$ in the ...

**2**

votes

**0**answers

225 views

### Finer properties of a sequence of harmonic functions

This was a question that arose for me when I was thinking about how one proves strong unique continuation for elliptic equations. I never could come up with a satisfactory answer.
Background:
When ...

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**0**answers

222 views

### Why is a smooth weak solution strong for stationary linear Stokes problem with zero-traction boundary condition?

Can anyone provide me with a reference giving details on how smooth generalized solutions of the stationary linear Stokes problem can be shown to be classical solutions when a zero-traction boundary ...

**1**

vote

**0**answers

62 views

### Gilbarg-Trudinger's book Theorem 4.13

I am reading Gilbarg-Trudinger's book "Elliptic Partial Differential Equations of Second Order". I do not understand the proof of Theorem 4.13.
Theorem 4.13 is a special case of Kellogg's theorem in ...

**1**

vote

**0**answers

85 views

### Asymptotics of “heat” semigroup

Consider a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary. Consider a second order elliptic operator $L$ on $L^2(\Omega)$, defined by either the Dirichlet or Neumann boundary ...

**1**

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**0**answers

60 views

### Energy inequalities for Sobolev spaces of negative integer

I asked this question in mathematics stackexchange and couldn't get an answer.
Let $\phi\in H^{s}$ such that the following energy inequality is true:
$$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| ...

**1**

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95 views

### Elliptic regularity Schauder estimates with Dirichlet/Neumann boundary conditions

Consider the linear elliptic equation $Lu = 0$, where $L$ is a second degree elliptic operator with smooth coefficients on a bounded domain $\overline{\Omega} \subset \mathbb{R}^n$, where $\Omega$ is ...

**1**

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**0**answers

88 views

### Size of the eigenfunction of Laplacian (reference request)

It is a classical Sobolev inequality that if $\phi$ is an eigenfunction of the Laplace-Beltrami operator on a $n$-dim compact Riemannian manifold $M$ with eigenvalue $\lambda$ then
...

**1**

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**0**answers

52 views

### Equivalence of two definitions of weak solution (subtlety with null sets)

Consider
$$y_t - \Delta y = f$$
$$y(0) = y_0$$
with zero boundary condition. Let $a(t,.,.)$ be the bilinear form associated to $-\Delta$. We have two definitions of weak solutions:
We have $y \in ...

**1**

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**0**answers

102 views

### $L^2$ bound on solution of PDE in terms of $L^2$ norm of initial value

Let $u \in H^1((0,T)\times S)$ be the unique solution of
$$u_{tt} + \Delta u =0$$
$$u|_{t=0}= u_0$$
$$u|_{t=T}=0$$
where $u_0 \in H^{\frac 12}(S)$ and $S$ is some Euclidean hypersurface without ...

**1**

vote

**0**answers

43 views

### sharp conditions characterizing the vanishing of scalar Jacobi fields

Let $T>0$, let some function $\kappa(t)$ smooth on $[0,T]$, and let $b$ the unique solution to the ODE $\ddot b + \kappa(t) b = 0$ with initial conditions $b(0)=0$ and $\dot b(0) = 1$.
I was ...

**1**

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**0**answers

26 views

### Recursive formula for symbol of resolvent on noncompact manifold

On a compact Riemannian manifold $(M,g)$ without boundary it was shown (by R. Seeley) how to define the complex power of an elliptic classical pseudodifferential operator $A$ of positive order $m$: ...

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50 views

### persistence of regularity for nonlinear Klein-Gordon equation

I have been reading the paper on nonlinear Klein-Gordon equation(NLKG) for initial data in modulation space: For detail please see the paper "Klein-Gordon Equations on Modulation Spaces (2014)" ...

**1**

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63 views

### maximum principle on compact manifolds with boundary

Let us consider the equation $Lu + f(u) = 0$ on a compact manifold $\overline{M} = M \cup \partial M$ with boundary, with Dirichlet boundary conditions. $L$ is a linear elliptic operator, and $f$ ...

**1**

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62 views

### Cauchy Problem and stochastic representation for discontinuous initial data

Where can I read more about the Cauchy problem, i.e. solutions to
$$ \frac{\partial u}{\partial t}+Lu=0 \text{ and } u(0,x)=f(x)$$
for some elliptic differential operator $L$ where $f$ is not ...

**1**

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**0**answers

73 views

### $H=W$ for weighted Sobolev spaces

Meyers and Serrin's $H=W$ is well known, but how does it generalize when we add weights?
Let's define $H^{m,p}(\mu_0,\dots,\mu_m)$ to be the completion of $C^\infty(\Omega)$ in the norm
...

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**0**answers

54 views

### Existence for special Dirichlet problem

I would like to know the following: Let $M$ be a smooth surface with connected boundary. Let $f: M \rightarrow \mathbb{R}^3$ be an embedding such that the boundary $\partial M$ of $M$ is mapped onto ...

**1**

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106 views

### Strichartz estimates for the wave equation

Strichartz estimates for the wave equation $\Box u=F $ with $u(0)=g_0$ and $\partial_t g(0)=g_1$ can be stated as
$$\Vert u\Vert_{L^q_tL^r_x}+\Vert u\Vert_{C^0_t\dot{H}_x^s}+\Vert\partial_t ...

**1**

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46 views

### Isolated singularities of harmonic mappings

Can a homeomorphic harmonic mapping $f=(u,v,w):\Omega\to \Omega'$ have isolated singular points. Here $\Delta f =0$, and singular point is a point with zero Jacobian. This will extend Lewy theorem for ...

**1**

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57 views

### Strong solution to parabolic equation without differentiability assumption on coefficient?

Consider on $(0,T)\times \Omega$, $\Omega$ a bounded domain
$$u_t(t,x) - a(u(t,x))\Delta u(t,x) = f(t,x)$$
$$u|_{\partial\Omega} = 0$$
where $a$ is real-valued and satisfies
$C_1 \leq a(r) \leq C_2$ ...

**1**

vote

**0**answers

116 views

### Harmonic extension of $L^\infty$ function is in $L^\infty$?

Let $u \in H^{\frac 12}(\Omega)$ with $\int_\Omega u = 0$ and consider the solution $v \in H^1(C)$ where $C=\Omega \times (0,\infty)$ of
$$-\Delta v(x,y) = 0$$
$$\partial_\nu v = 0$$
$$v(x,0) = ...

**1**

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**0**answers

36 views

### Finite elements $W^{1,\infty}$ error estimates

Are there finite element method setups that provide error estimates in the $W^{1,\infty}$ norm (i.e., bounds on $\|u'_h - u'\|_\infty$)? Which families of elements can be used for implementing them?

**1**

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191 views

### A Lie algebra associated with a one dimensional foliation

A non vanishing vector field $X$ on a manifold is called "well behaved" if for every non vanishing smooth function $f$ we have $$C(X)\simeq C(fX)$$ This means that the centralizer Lie algebras ...

**1**

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42 views

### Density of restrictions of $p$-harmonic functions on a hypersurface

Let $\omega,\Omega\subset\mathbb R^n$, $n\geq2$, be bounded smooth domains so that $\bar\omega\subset\Omega$.
Let $1<p<\infty$.
Define the boundary space $B=W^{1,p}(\omega)/W^{1,p}_0(\omega)$; ...

**1**

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**0**answers

132 views

### Help in understanding “Local well-posedness for the Maxwell-Schrodinger system”

Is there someone who knows the following paper
"Local well-posedness for the Maxwell-Schrodinger system" by M.Nakamura and T.Wada.
I'm trying to study it but I've some doubts. In particular I'm not ...

**1**

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**0**answers

61 views

### Question on the local existence theory for the classical solution for the incompressible fluid dynamics equation

For the incompressible fluid dynamics system
\begin{equation}
\begin{split}
&\nabla\cdot v=0,\\
&\dfrac{\partial v}{\partial t}+v\cdot \nabla v+\nabla p=A(S,D)v,
\end{split}
\end{equation}
...

**1**

vote

**0**answers

76 views

### variation norm of a Fourier transform

Motivated by certain uniform estimate in oscillatory integrals, I am now trying to calculate the Fourier transform of the function ${\large e^{i|t|^{\epsilon}}/t}$ on $\mathbb{R}$, where $\epsilon\in ...

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78 views

### A bilinear estimate in Lp space

Let $\varphi(D)$ be a Fourier multiplier with symbol $\varphi(\xi) = \xi/(1+|\xi|^2)$. It's easy to prove that
\begin{equation}
\|\varphi(D)u^2\|_{H^s(R)}\lesssim \|u\|^2_{H^s(R)} \quad (*)
...

**1**

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84 views

### Free Endpoint of Minimization Problem

Consider the following minimization problem $$\inf \left\{ \int\limits_{-\infty}^0 \left[ (\psi')^2 + m(y)(\psi - F)^2 \right]\; : \; \psi \in H^1(\left(-\infty,0\right]) \right\}$$ where $m(y) > ...

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40 views

### Introduction to free boundary problems (that are not Stefan problems)

Could someone recommend some notes/papers that deal with existence/regularity of free boundary problems arising from parabolic equations (excluding Stefan type equations)?
I am thinking of eg. ...

**1**

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**0**answers

25 views

### Is it classical that the solution to an hyperbolic equation equation is Lipschitz -continuous $[0,\infty)\to L^1(\mathbb{R})$?

Recently I am reading a paper "Global solution and smoothing effect for a non-local regularization of a hyperbolic equation" published on J.E.E, 2004. In the proof, the authors write "It is classical ...

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49 views

### decomposition of tempered distributions by entire analytic functions

Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1~~\text{if}~|\xi|\leq 1\}$$
Let $j\in \mathbb N$ ...

**1**

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71 views

### Measurability of solution of diffusion equation in sub sigma algebra

I want to solve the following problem:
Get $\omega \in \Omega \subset \mathbb{R}$, $x \in D \subset \mathbb{R}^2$ and $0<a_i\leq a(.,.)\leq a_x<\infty$.
Let $a( x;. )$ and $f(x;.)$ be ...

**1**

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114 views

### How does perturbation method guarantee its solution for the perturbed pde $\Delta u + \epsilon u^2 =0$

[This may not be a research-level question; if it violates any term of this website, I will delete it right away]
My question is quite simple: Suppose we are given a PDE of with a boundary condition
...

**1**

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**0**answers

54 views

### Bogomol’nyi’s Formula for the Critical Action

I'm studying Aigner's paper 'Existence of the Ginzburg-Landau Vortex Number' (2001) and I have some difficulties to prove the equality (3.1) , which is
...

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137 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 ...

**1**

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**0**answers

78 views

### Uniform bounds for a coupled parabolic system of PDE (linear)

Let $V=H^1(\Omega)$ and $H=L^2(\Omega)$ where $\Omega$ is a compact Riemannian manifold. Define $W = \{ w \in L^2(0,T;V) : w_t \in L^2(0,T;V^*)\}$.
Consider the system, with $u^\epsilon, v^\epsilon ...

**1**

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149 views

### numerical method (implicit , backward difference or forward difference) for nonlinear pde

$\newcommand{\lbar}{\underline{\lambda}}$
In this linear PDE:
\begin{cases}
B_t+b^Q(r,t)B_r+\frac{1}{2}d^2(r,t)B_{rr}+(\mu(\lambda,t)+\alpha \sigma (t))(\lambda -\lbar)B_{\lambda} \\ ...