# Tagged Questions

**1**

vote

**1**answer

217 views

### In which way is this a linearization of the Gross-Pitaevskii-Equation?

In their paper [1] (full text at [2]) Bethuel et al on page 249 (bottom) linearize the moving frame Gross-Pitaevskii-Equation
$0=-ic \partial_{x_1} \widetilde{v} - \Delta \widetilde{v} - ...

**0**

votes

**1**answer

76 views

### Integral of a harmonic function on a manifold with two non-parabolic ends

Let M be a complete Riemannian manifold.Suppose there are two non-parabolic ends on M with respect to $M\backslash {B_p}\left( {{R_0}} \right)$Then there is a harmonic function f on M.Is it right that ...

**2**

votes

**0**answers

86 views

### P-laplacian equation

Hi guys,
in what sense the p-Laplacian is degenerate for p greater than 2 and singular for p smaller than 2 ?
Thank you!

**10**

votes

**1**answer

533 views

### Do eigenfunctions of elliptic operator form basis of $H^k(M)$?

We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$ and $L^2(M)$.
If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ ...

**2**

votes

**0**answers

182 views

### Extension divergence-free, curl-converging vector field

Hi.
Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole ...

**22**

votes

**1**answer

1k views

### What goes wrong for the Sobolev embeddings at $k=n/p$?

For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the celebrated Sobolev inequalities:
If $k < n/p$ then $u\in L^q(U)$ for $q$ satisfying ...

**1**

vote

**0**answers

81 views

### Orthogonal projection of discontinuous piecewise polynomial space in energy scalar product

Let $I = [0,1]$ be the unit interval Let $I$ be partioned into $n$ closed subintervals $(I_j)_J$, each of length $1/n$.
Let $X_{DC} = \{ v \in L^2[0,1] | 1 \leq j \leq n : v_{|I_j} \in \mathcal P_1( ...

**1**

vote

**2**answers

239 views

### Regularity of parabolic equation; divergence free drift

I am curious about weak solutions of the parabolic problem
$ u_t - \Delta u + a(x,t) \cdot \nabla u = f(x,t)$ in $ \Omega \times (0,\infty)$ with $ u=0$ on $ \partial \Omega$. Here $a(x,t)$ is ...

**1**

vote

**2**answers

179 views

### Stokes problem; naive question on the regularity of pressure term

I am attempting to recall some basic knowledge related to Stokes' problem.
In particular I am following along in Evans PDE book in section 8.4.
So lets assume that
$-\Delta u + \nabla P = f $ in $ ...

**1**

vote

**0**answers

163 views

### multivalued solution of a equation

Definition: A scalar k-th order differential equation on a smooth manifold $M$ ,
is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $
for $\left | \sigma \right |\leqslant ...

**0**

votes

**1**answer

132 views

### Higher dimensional analogue of Kellog's theorem? (Holder continuity of solution to Dirichlet problem with Holder continuous boundary data)

Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the following
Theorem: ...

**7**

votes

**1**answer

296 views

### Linearization instability and singular points of algebraic varieties

In a well known 1973 paper, Fischer and Marsden pointed out (with similar, contemporary remarks made in the physics literature by Brill and Deser) that the space of solutions of some non-linear ...

**0**

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**0**answers

143 views

### Splitting the action of functionals in duals of Sobolev spaces

Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...

**3**

votes

**1**answer

154 views

### Coarea formula in a subelliptic context

Consider smooth vector fields $X_1,..,X_k$ in ${\mathbb R}^n$, satisfying the H\"ormander condition, i.e. for all $x$, the Lie algebra generated by $X_1(x),...,X_k(x)$ is ${\mathbb R}^n$. Do you know ...

**2**

votes

**1**answer

150 views

### How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$?

How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$?

**0**

votes

**1**answer

314 views

### Existence of a function

[also asked here http://math.stackexchange.com/questions/307197]
All arguments are in $\mathbb{R}^3$.
Suppose $n(x)$ is a smooth function where $\mathbf{supp}(n(x)-1)$ is a compact set $\Omega$. ...

**4**

votes

**1**answer

823 views

### Nash's paper on parabolic equations.

I am currently studying the paper "CONTINUITY OF SOLUTIONS OF PARABOLIC AND
ELLIPTIC EQUATIONS" by John Nash (cf. American Journal of Mathematics, Vol. 80, 1958). The author there establishes some a ...

**1**

vote

**1**answer

97 views

### Boundedness of a given boundary value problem.

I've been given the following BVP:
\begin{align*}
-\Delta u = u- u^3,\: x\in \Omega
\end{align*}\begin{align}
u = 0,\: x\in \partial \Omega
\end{align}
where $\Omega\subset \mathbb{R}^N$ is bounded.
...

**2**

votes

**0**answers

198 views

### Convergence rate of iterated nonlinear equations?

For $i=1, \dots, n$ ($n$ could be large) we have variables $x_i$ and $y_i$ relating to probability bounds s.t. $x_i, y_i \geq 0, x_i+y_i \leq 1 \; \forall i$. Each $i$ has a constant $\theta_i$, and ...

**4**

votes

**1**answer

259 views

### Is there a good way to estimate the Fourier transform of $\frac{1}{\lambda-iP(\xi)}$

Assume that P is a real valued strong elliptic polynomial, then what do we know about the following
$$
K(\lambda,x)=\int{\frac{e^{ix\xi}}{\lambda-iP(\xi)}}d\xi,\quad \lambda\in \mathbb{R}\0
$$
The ...

**0**

votes

**1**answer

213 views

### Quantitative Global Schauder Estimates/Hölder Regularity

Consider the linear second order elliptic Dirichlet problem
$$-\nabla\cdot (a\nabla u)\quad u=0 \text{ on }\partial\Omega$$
Condtion 1:$\Lambda |\xi|^2\geq\sum a_{i,j}(x) \xi_i \xi_j \geq \lambda ...

**0**

votes

**0**answers

278 views

### Eigenfunctions of the laplace on the 2-sphere with conformal metric induced by schwarzschild

Hi,
it's well known that the coordinates $x_1$, $x_2$, $x_3$ are the first three eigenfunctions with positiv eigenvalues ($=2/r^2$) of the negative laplace-beltrami operator ${-}\triangle$ on the ...

**3**

votes

**1**answer

286 views

### on an inequality of Brezis-Lieb

In their 1983 JFA paper Brezis and Lieb have shown, among many other things, a Poincaré-type inequality: in the case of a harmonic function $f$ on a bounded domain $\Omega$, their inequality ((3.14) ...

**0**

votes

**0**answers

205 views

### “Integration by parts” formula for functionals

We know that for a Hilbert triple $V \subset H \subset V^{'}$, if we have $u, v \in L^2(0,T;V)$ with $u',v' \in L^2(0,T;V')$
then
$$\frac{d}{dt}(u(t), v(t))_H = u'(t)(v(t)) + v'(t)(u(t))$$
where the ...

**2**

votes

**1**answer

621 views

### Norm of differential operator between Sobolev spaces

It is easy to check that the differential operator $\partial^a$ (where $\alpha\in \mathbb{N}_0^n$) is continuous between the Sobolev spaces (with usual norms)
$W^{m,p}(U)\to W^{m-|\alpha|,p}(U)$, ...

**1**

vote

**0**answers

123 views

### Spectrum of Combinatorial Laplacian

The spectrum of the combinatorial laplacian is well understood for a square lattice. What about for other lattices?
In particular:
Let $ f: \mathbb{Z}^2 \rightarrow \mathbb{R} $. The usual ...

**4**

votes

**0**answers

191 views

### Viscosity solution of the PDE

Let $\Omega$ be bounded domain, $u=0$ on $\delta\Omega$ and
$$|Du|-f(x,u)=0$$
where $f\ge 0$ and $f$ is strictly monotone for fixed $x.$ I am looking for the reference to show that it has unique ...

**3**

votes

**0**answers

255 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

270 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 ...

**4**

votes

**1**answer

299 views

### Robin-Laplacian in unbounded domains

Let $\Omega\subset \mathbb R^n$ be an open domain and $\tau>0$. Consider the following boundary value problem
$-\Delta v=f $ in $\Omega$, $\partial_\nu v+\tau v=g$ on $\partial\Omega$.
If ...

**2**

votes

**1**answer

249 views

### Solvability for constant-coefficient partial differential operators

Let $\mathcal{S}$ denote the space of Schwartz functions on $\mathbb{R}^n$, and $\mathcal{S}'$ the space of tempered distributions. Let $L$ denote a linear, constant-coefficient, partial differential ...

**6**

votes

**0**answers

338 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

**0**answers

198 views

### Reference request: Anisotropic Sobolev spaces

Hello,
I am interested in what is known about anisotropic Sobolev spaces, by which I mean spaces of functions satisfying
$ \| f \|_p < \infty, \|Df \|_q < \infty, $
where $p \ne q$ (as ...

**2**

votes

**0**answers

69 views

### Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform II

This is a modification of a previous question.
Question: Suppose $a(s)\in C^\infty([0,1])$ and $H(s,x)\in C^\infty([0,1]\times [0,1])$ with $H(s,x)>0$, $\forall s,x\in [0,1]$. Suppose,
...

**9**

votes

**3**answers

393 views

### Boundedness of the derivative of the trace of an H^1 function

As a research preface, this question is linked to a problem of increasing magnetism in Ginzburg-Landau equations that I have distilled for the purpose of getting to the bottom of this technical ...

**0**

votes

**0**answers

142 views

### Looking for higher order Sobolev inequality

Hello,
On a compact (without boundary) Riemannian manifold (eg. some surface in $\mathbb{R}^n$), I'm looking for a result like
$$\lVert \nabla u\rVert_{L^2}^2 \leq \epsilon\lVert ...

**1**

vote

**1**answer

259 views

### A 'conjecture' on critical elliptic pde

I conjecture the following.
Let $\Omega=\mathbb{R}^3-\overline{B_1(0)}$. Define
$$E_{\Omega}(u)=\frac{1}{2}\int_{\Omega}|\nabla u|^2dx-\frac{1}{6}\int_{\Omega}|u|^6dx.$$
$E_{\mathbb{R}^3}$ is defined ...

**12**

votes

**1**answer

534 views

### Symbols of elliptic operators

First let me state the problem, then I'll explain its origin and finally, I'll ask the main question..
Problem S. Fix a positive integer $n$. Find all the pairs $(V, S)$, whith the following ...

**1**

vote

**0**answers

94 views

### A critical elliptic PDE

I am considering the problem $-\Delta u=|u|^4u$, $x\in \Omega\subset \mathbb{R}^3$, $u|_{\partial \Omega}=0$. Where $\Omega$ is a unbounded domain. Some special case like $\Omega=\mathbb{R}^3-B_1(0)$, ...

**2**

votes

**2**answers

195 views

### The logarithmic fast diffusion equation in one space variable with periodic boundary conditions.

I need to know about this non-linear logarithmic fast diffusion equation for a function $u(x,t)$ of one space variable $x$ and time $t$:
$$ u_t = (\ln u)_{xx}$$
which is to run on an interval $ a \leq ...

**2**

votes

**0**answers

109 views

### Best constant of Gagliardo-Nirenberg inequality in exterier domain

In $\mathbb{R}^N$, we know that the best constant of Gagliardo-Nirenberg is characterized by the solution $Q$ of $-\Delta u+u=|u|^2u$ with minimal mass. One have
$||u||_4^4\leq C||u||_2||\nabla ...

**2**

votes

**1**answer

149 views

### The distribution of roots of elliptic polynomial

If $p(x)$ is an $n$ variables polynomial of even degree with complex coefficients which satisfies the strong elliptic condition, that is, Re$p(x) \ge C|x|^{2m}$ for any $x \in \mathbb R^n$ where $2m$ ...

**1**

vote

**1**answer

349 views

### Existence of solution for this parabolic PDE

The parabolic PDE
$$\langle u', v \rangle + a(u,v) = \langle f, v \rangle$$
has a unique solution $u \in L^2(0,T; H^1)$ with $u' \in L^2(0,T;H^{-1})$ if $a$ is a bounded and coercive bilinear form ...

**1**

vote

**2**answers

248 views

### Are $\lVert \Delta u \rVert_{L^2(S)}$ and $\lVert u \rVert_{H^2(S)}$ equivalent norms on a compact manifold?

Hi,
I am looking for the result:
$$\text{The norm} \quad \lVert \Delta u \rVert_{L^2(S)} \quad \text{is equivalent to} \quad \lVert u \rVert_{H^2(S)}$$
for scalar functions $u \in H^2(S)$, where $S$ ...

**4**

votes

**1**answer

155 views

### variation of the obstacle in the obstacle problem

Suppose $D \subset \Bbb C$ with smooth boundary. Let $f \in C^{1,1}(D)$. Let $\varphi$ be the supremum of all members in the set
$$\lbrace g \in C^{\infty}(\overline{D})| g \ is \ subharmonic \ and ...

**2**

votes

**2**answers

284 views

### how many nonparabolic ends guarantee a nonconstant harmonic function on Riemannian manifold?

M is a Riemannian manifold.An end E is said to be a non-parabolic end if it admits a positive Green's function with Neumann boundary conditon on E.Otherwise,it is a parabolic end.If M has more than ...

**1**

vote

**1**answer

476 views

### weak derivative and continuous function

Let $\Omega \subset \mathbb{R}^n$ be a compact smooth hypersurface. Suppose $\varphi \in C_c^\infty(0,T; H^1(\Omega))$ is a $H^1(\Omega)$-valued test function (so $\varphi(t) \in H^1(\Omega)$ for each ...

**9**

votes

**3**answers

562 views

### Space of sections of a fibre bundle with non-compact base space

Let $\pi: E \rightarrow M$ be a fiber bundle over the manifold M and denote by $\Gamma(E)$ the space of smooth sections of $E$.
For compact $M$ it is well known (Hamilton 1982, Part II Corollary ...

**1**

vote

**2**answers

281 views

### Fourier transform of function on compact set and Sobolev norm equivalence

Hi all. My question on M.SE is unanswered (http://math.stackexchange.com/questions/254970/fourier-transform-of-function-defined-on-subset-of-mathbbrn) so I want to post it here. I changed it slightly.
...

**7**

votes

**3**answers

480 views

### abstract evolution equations

Hi
Whenever I read a book on evolution equations, they set up, say the parabolic PDE
$$\dot{y} = Ay + f$$
in abstract function spaces (eg. $L^2(0,T;V)$ and $L^2(0,T;V^*)$). In examples, they always ...