# Tagged Questions

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132 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

**0**answers

337 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
$$
{\|u - \frac{1}{\Omega} \int_{\Omega} u\|}_{L^2}\leq C
{\|\nabla u \|}
$$
...

**2**

votes

**0**answers

365 views

### Convolution Estimates on a Smooth Manifold

Suppose $f,g$ are $a$-Hoelder continuous real-valued functions on some domain $\Omega \subset \mathbb{R}^n$ satisfying
$$
\|f\|_{C^{0,a}(\bar\Omega)},\|g\|_{C^{0,a}(\bar\Omega)}<\infty.
$$
Then ...

**2**

votes

**0**answers

216 views

### Pitfalls when generalizing the heat kernel of a Riemannian metric

Suppose $M$ is a Riemannian manifold with some compact quotient under isometries.
Associated with the Riemannian metric one has the Laplace-Beltrami operator $\Delta$ and the heat kernel $p(t,x,y)$ ...

**2**

votes

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155 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**

votes

**0**answers

85 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

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163 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 ...

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**0**answers

331 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

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196 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 ...

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253 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

**0**answers

697 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**

votes

**0**answers

150 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**

votes

**0**answers

324 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**

votes

**0**answers

289 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**

votes

**0**answers

249 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**

votes

**0**answers

310 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**

votes

**0**answers

303 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

235 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

229 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

64 views

### Applications of the Weak and Weak$^*$ topologies to PDEs?

Chapter $3$ of Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis constructs and explains the Weak and Weak$^*$ topologies over a Banach Space $E$.
The most ...

**1**

vote

**0**answers

38 views

### Hyperbolic PDE from total derivative?

Given a density function $p(t, \boldsymbol{x})$, where $t$ is time and the vector $\boldsymbol{x}$ represents a point in $n$ dimensional space, a hyperbolic PDE describing the time evolution of the ...

**1**

vote

**0**answers

108 views

### The Yamabe problem and $\phi^4$ scalar field theory?

The other day I happened to be browsing this page on wikipedia: https://en.wikipedia.org/wiki/Mass_gap
In the middle of the page is the equation $$\square\phi+\lambda\phi^3=0$$ where $\square$ is the ...

**1**

vote

**0**answers

71 views

### Maximum Principle for Elliptic Equation with Mixed Boundary Conditions

Given the equation
$$ -\nabla\cdot(a_{ij}\nabla u)=0$$
with mixed boundary conditions
$$ u|_{\partial\Omega_D}=u_D,\quad \frac{\partial u}{\partial \nu}|_{\partial\Omega_N}=0$$ and appropriate ...

**1**

vote

**0**answers

58 views

### Fluid dynamics of a rotating liquid droplet

I'm looking for an analytical solution of the Navier-Stokes equation with the following boundary conditions: a liquid is held inside a spherical shell, which is rotating at a constant rate, and is ...

**1**

vote

**0**answers

53 views

### Motivation for some operators in the dyadic model of Navier Stokes equation

What's the motivation for cascade operator $C_{u}, C_{d}$ (and then the dyadic version of operator $B=(u\cdot \nabla)u$), which is
$(C_{u}(u,v))_{Q} = 2^{\frac{5}{2}j}u_{\tilde{Q}}v_{\tilde{Q}}$,
...

**1**

vote

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37 views

### The decay rate of the spectrum of the Gaussian kernel on compact manifolds

It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...

**1**

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66 views

### $C^{1,2}$ regularity of (weak) solutions to the heat equation

Let $\Omega$ be a bounded Lipschitz domain (smoother if needed), and consider the heat equation
$$u_t - \Delta u = 0$$
$$\frac{\partial u(t,x)}{\partial \nu(x)} = a(t,x) - b(t,x)u(t,x)$$
$$u(0) = ...

**1**

vote

**0**answers

66 views

### How is the duality pairing of $H^{1/2}$ and $H^{-1/2}$ defined on a subset of the boundary?

Let $\Omega\in \mathbb{R}^d$,for $d\in \{2,3\}$ be a bounded polyhedral set with $n$ boundary faces labeled $\{e_i\}_{i=1}^n$.
Let $\vec{q}\in H^{\mathrm{div}}(\Omega).$ Given $u\in ...

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vote

**0**answers

68 views

### An H2 estimate for Helmholtz equation

How to show the following statement?
Let $\Omega$ be a bounded Lipschitz convex domain. If $u$ satisfies the following equation,
$$
-\Delta u - k^2 u = f \quad\mbox{ in }\Omega \\ \nabla u ...

**1**

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55 views

### Characteristic field

On page 423 (page 9 of PDF) of the paper by Lions and Perthame, the authors write as equation (39) the Vlasov equation as
$$\frac{\partial f}{\partial t} + v \cdot \nabla_x f + F \cdot \nabla_v f = ...

**1**

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**0**answers

32 views

### About the “method of lines”: when are such solutions good approximations for **all** future time?

This question is about approximate solutions to some classes of PDEs obtained using the "method of lines".
For example, for an initial-value problem given by a PDE on a circle, one can choose $n$ ...

**1**

vote

**0**answers

111 views

### Equation of the form $\overline{\partial}_{J}f=g$ made holomorphic

In section 2.6 of the book "Holomorphic curves in symplectic geometry" by Audin and Lafontaine there is explained when one can transform a perturbed holomorphic curve in a holomorphic curve. I tried ...

**1**

vote

**0**answers

28 views

### Vector fields for volumetric-deviatoric decomposition

The strain tensor $\epsilon(u) = \frac12 (\nabla u + (\nabla u)^T)$ in linear elasticity can be decomposed additively into volumetric and deviatoric strains
\begin{gather*}
\epsilon_D(u) &= ...

**1**

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61 views

### Properties of a Sobolev bound

I am interested in computing
$$
A:=\inf_{f\in L^{2}(\mathbb{R}^3)}\frac{||K^{\frac{1}{4}}f||_2^2}{||f||_{\frac{5}{2}}^2}
$$
where $K:=-\Delta+1$. We call $f_c$ the function that saturates the bound.
...

**1**

vote

**0**answers

58 views

### Perturbation in Besov space

$\|f\|_{B^{0}_{p,p}}=(\sum_{j\geq -1} \|\Delta_j f\|_p^p)^{1/p}$ is the Besov norm of $f$.
Here the Fourier transform of $\Delta_jf~(j\geq 0)$ is $\psi(2^{-j}\xi)\hat{f}(\xi)$ and $\psi$ is a smooth ...

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**0**answers

31 views

### Biharmonic Equation in a Rectangle with Some Uncommon Boundary Conditions

I asked this question on Mathematics network but it didn't receive any answers. So I assume it is just beyond the classic things in PDEs and I decided to ask it here too.
Consider the following ...

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**0**answers

51 views

### Suggestion for books in Pertubation theory with an emphasis on the theory

As the title suggest I am looking for another good coverage of the theory of Pertubation theory.
Currently I am working through Murodock's book: Pertubations: Theory and Methods.
But I am rest assure ...

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**0**answers

75 views

### Wave-like equation with 1st order time derivative and non-constant coefficients

We start with the following recurrence relation for complex coefficients $c_{n,m}$:
$$i\dot{c}_{n,m}(t) = \sqrt{(n+1)(n+2)(m-1)m}c_{n-2,m+2} + \sqrt{n(n-1)(m+1)(m+2)}c_{n+2,m-2}$$
where ...

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**0**answers

69 views

### Boundary regularity of solution to partial differential equation

I am conducting research on partial differential equations and I need a short-time existence result from the literature which I can not find at the moment. More precisely I would like to know the ...

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**0**answers

81 views

### Heat semigroup ultracontractive?

Let $g(x,t)= \frac{1}{(4 \pi t)^{\frac{n}{2}}}e^{\frac{-|x|^2}{4t}}$ be the heat kernel on $\mathbb{R}^{n}.$ Is the standard definition now to say that this heat-semigroup $T(t)(f):=g *f(.,t)$ is ...

**1**

vote

**0**answers

63 views

### A $\mathcal{C}^1$ domain and Hausdorff dimension estimate

Let us consider an open connected domain $E\subset \mathbb{R}^N$ and fix a point $x_0\in\partial E$ on its boundary.
Suppose now, that there exists $R>0$ such that the set $\partial E \cap ...

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139 views

### On the differentiability of a certain map from $ (0,\infty) $ to $ \Bbb{R} $

This problem arose from my study of energy-conservation for non-linear Schrödinger equations. Suppose that we have the following data:
$ u \in C^{1} \! \left( (0,\infty),{L^{2}}(\Bbb{R}^{n}) \right) ...

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39 views

### Fundamental gap for Neumann BVP with potential

I am sure this is extremely well known but I have been digging a bit and I can't find what I need. Consider $B$ to be the unit ball in $ \mathbb R^N$ and consider the eigenvalue problem
...

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49 views

### method for global existance for the NLS

We consider the nonlinar Schr\"odinger equation(NLS):
$$i\frac{\partial u}{\partial t}+\Delta_{x}u +\lambda |u|^{2k}u =0, \ u(x, 0)= u_{0}(x);$$
where $\lambda \in \mathbb R, \ k \in \mathbb N,$ ...

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61 views

### Regularity on Neumann problem on polygonal domain

I asked a similar question before but didn't get any responses. So I will attempt again (the prior question was regarding Holder continuity).
Let $ \Omega$ denote a cube in $ R^n$ and consider ...

**1**

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**0**answers

69 views

### Asymptotics of a elliptic pde when exponent gets large

I am interested in the following pde
$$ -\Delta w_p + \left( \frac{1}{p-2} +1 \right) \frac{ | \nabla w_p|^2}{w_p} + \epsilon(p) \left( \frac{1}{w_p} \right)^{(p-2)} = (p-2) w_p $$ in the unit ball ...

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26 views

### Regularity of a flux induced by a potential

Take
$\Omega\subset R^n$ with smooth boundary (take a ball for example)
a function $f\in L^{\infty}(\Omega)$ with support strictly contained in $\Omega$ and with $\int _{\Omega} f \; dx=0$
a ...

**1**

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74 views

### One-parameter group of unitary operators and Core

Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...

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97 views

### A Question about compactness of an embedding into $L^p$ spaces

Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says
For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have
$$ \Lambda \int_{\Omega} ...

**1**

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80 views

### Existence of at least one positive solution for semilinear biharmonic equation with critical exponent

Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow
\begin{cases} \Delta^2u=\lambda \dfrac{u}{|x|^4}+u^{p} & ...