# Tagged Questions

Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

**8**

votes

**1**answer

296 views

### Are conformal maps between Riemannian manifolds real-analytic?

This is a cross-post.
Let $M,N$ be oriented smooth ($C^{\infty}$) $n$-dimensional Riemannian manifolds, and let $f:M \to N$ be a smooth orientation-preserving weakly* conformal map.
Do there ...

**2**

votes

**0**answers

67 views

### Ricci flow with surgery without the “no locally separating $\Bbb RP^2$” assumption

In many places, Ricci flow with surgery is done with orientable manifolds. Morgan and Tian do not require orientability, but instead they impose the condition that $M^3$ have no embedded $\Bbb RP^2$ ...

**3**

votes

**0**answers

47 views

### Ratio of solutions to two heat equations

Let $u(x,t)$ and $v(x,t)$ respectively solve the two one-dimensional heat equations with different (real) diffusion coefficients on the same domain $D$ and the same initial & boundary conditions, ...

**4**

votes

**0**answers

37 views

### Global existence of strong solutions for 1D Boltzmann equation

I have recently get interested into kinetic theory equation and I would like to know more about it.
In particular, I am reading the paper "Strong solutions of the Boltzmann equation in one spatial ...

**1**

vote

**2**answers

39 views

### Unboundedness of the Sobolev norm of a sequence : Does it follow from Sobolev embedding?

Let $\{f_n\}$ be a sequence of functions that are continuous and lying in $H^k(\mathbb{R}^m)$. Assuming $k>\frac{m}{2}$, and if $f_n \to f$ pointwise, where $f\in H^k(\mathbb{R}^m)$, such that $f$ ...

**0**

votes

**0**answers

63 views

### any hope of $C^{2,\alpha}$ regularity for linear problem

Let $B_1$ denote the unit ball in $ R^N$ (say $N \ge 3$) and consider
$$ \Delta \phi + \gamma \sum_{i,j=1}^N \frac{x_i x_j}{|x|^2} \phi_{x_i x_j} = f $$ in $B_1$ with $ \phi=0$ on $ \partial B_1$. ...

**7**

votes

**2**answers

554 views

### Eigenvalues of Laplace-Beltrami on half sphere

Let $ \Delta_\theta$ denote the Laplace-Beltrami operator on $S^{N-1}$. The eigenvalues of this are well known. I assume the same is the case of this operator on the upperhalf sphere; say $ S^{N-...

**0**

votes

**0**answers

25 views

### regularity of function from spherical representation

I asked a question regarding the linear operator a few days ago here is this explicit linear operator hypo-elliptic .
The operator in question was $ L(\phi)=\Delta \phi(x) + \gamma \phi_{rr}$. We ...

**1**

vote

**0**answers

48 views

### Using semigroup theory for nonautonomous semilinear equations

We have the abstract evolution equation $$U^{\prime}=A(t)U+F(U), \quad U(0) = U_0.$$ If the operator $A$ is independent of time, we can get local existence of solutions by proving that $A$ is the ...

**1**

vote

**0**answers

35 views

### Error estimates in the $L^2$ norm of conforming FEM about Poisson’s equation with mixed boundary conditions

Consider Poisson’s equation
$$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$
with following mixed boundary cconditions
$$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$
$$\frac{{\...

**3**

votes

**2**answers

202 views

+50

### Proof of Littman-Stampacchia-Weinberger theorem on the fundamental solution for elliptic PDEs

Where can I find a (readable and self-contained) proof of the following result?
Let $\Omega$ be a Lipschitz domain of $\mathbb{R}^n$, with $B(0,1) \subset \Omega$. Let $u$ be the solution of $$-\...

**2**

votes

**0**answers

106 views

### DeGiorgi oscillation lemma

Where can I find a proof of the following result?
Let $u$ be a subsolution of $$\mathrm{div}(A(x)\nabla u) = 0,$$
where $A$ is bounded, measurable and uniformly elliptic ($C^{-1}\mathrm{Id} \le A(...

**0**

votes

**0**answers

52 views

### $L^{\infty}-L^{\infty}$ estimates on the Schrödinger evolution

A classical estimate for solutions to Schrödinger equations are $L^1-L^{\infty}$ estimates, also known as dispersive estimates.
I wonder whether there are also $L^{\infty}-L^{\infty}$ estimates?
...

**0**

votes

**0**answers

48 views

### is this explicit linear operator hypo-elliptic

Consider an operator of the form
$$L(\phi):=\Delta \phi + \gamma \phi_{rr}$$ here the $r$ denotes derivative with respect to the radial variable (we are in $ R^N$ say where $N \ge 3$).
I am ...

**4**

votes

**1**answer

110 views

### compact embedding for Sobolev spaces

The Sobolev embedding $H_{0, rad}^{1}(\Omega) \hookrightarrow L^{p+1}(\Omega)$ is compact for all $p>1$ where $\Omega= \{x\in \mathbb R^N: 0<a<|x|<b\}.$
Is it possible to determine the ...

**5**

votes

**2**answers

200 views

### Making the Fourier transform quantitative

I am undergraduate Physics student and understand that this is a professional mathematics forum. But due to perhaps broader interest, I hope this question is suitable for this website.
I understand ...

**3**

votes

**0**answers

60 views

### Fractional embedding inequality with $L^{\infty}$ norm

Here we consider the fractional Sobolev spaces and suppose $u$ is a vector function in $\mathbb R^2$. For $q>2$, is the following always true? $$\Vert Du \Vert_{L^{\infty}(\mathbb R^2)} \leq C\Vert ...

**0**

votes

**0**answers

38 views

### Localizing a Clebsch-Gordan expansion around one representation

For the Lie group $\mathrm{SU}(2)$ the irreducible representations $\pi_m$ are labelled by non-negative integers $m$ and have dimension $(m+1)$. By the Peter-Weyl theorem, they form a basis for $L^2(\...

**3**

votes

**1**answer

109 views

### Global regularity for Neumann problem

Let $\Omega\subset \mathbb{R}^d $ be a bounded open subset ($d\in \mathbb{N}$) and denote $\partial\Omega$ its boundary which we assume to be Lipschitz. The classical inhomogeneous Neumann problem ...

**5**

votes

**2**answers

214 views

### Bounded deformation vs bounded variation

Let $BV(\mathbb R^n; \mathbb R^n)$ be the space of (vector-valued) functions of bounded variation and let $BD(\mathbb R^n;\mathbb R^n)$ the space of functions with bounded deformation. They are made ...

**3**

votes

**3**answers

169 views

### ODE with Bessel decay

This is related to my previous question, but it is probably less scary and an expert in using Mathematica could figure out an answer easily.
I would like to estimate the asymptotic behaviour of the ...

**2**

votes

**0**answers

44 views

### Smooth Cauchy problem on a cylindrical manifold (or how to define the exponential of a differential operator)

Let $M$ be a manifold, $E \rightarrow M$ be a real or complex smooth vector bundle, and $D: \Gamma_c(M,E) \rightarrow \Gamma_c(M,E)$ be a (first order if necessary) differential operator on smooth ...

**1**

vote

**0**answers

59 views

### Degenerate Monge-Ampere equation on a bounded domain with $C^{2,1}$ boundary

In the paper by Guan Pengfei: "C^2 a priori estimates for degenerate Monge-Ampere equations" https://projecteuclid.org/euclid.dmj/1077242669
Prof. P. Guan proved in Theorem 1 that the degenerate Monge-...

**1**

vote

**0**answers

101 views

### Estimate of $\Vert \nabla u \Vert_{L^{\infty}(\Omega)}$ of Navier Stokes equations

My question is how to estimate the term $\Vert \nabla u \Vert_{L^{\infty}(\Omega)}$. Here we consider the 2D incompressible Navier Stokes equations:$$u_t -\Delta u+u\cdot \nabla u+\nabla p=f$$ and $$\...

**2**

votes

**0**answers

90 views

### Minimizer of certain functional is strong $\Lambda$-minimizer

I'm interested in the quantitative isoperimetric inequality. I am currently reading https://arxiv.org/pdf/1007.3899.pdf and I have some questions regarding lemma 3.5 which states a certain situation ...

**2**

votes

**0**answers

37 views

### Bessel decay for nonhomogeneous PDE

I'm interested in the following nonhomogeneous PDE
$$ (\Delta-k^{2})u=-g $$
on the upper-half plane with smooth and integrable Dirichlet boundary condition, where $g$ is a smooth positive function ...

**2**

votes

**0**answers

76 views

### Inequality concerning BV norm

Let $u(x) \in L^1( \mathbb{R}^n) \cap BV(\mathbb{R}^n)$ and let $\rho\ge 0$ be the standard mollifier on $\mathbb{R}^d$, supported in unit ball with $\int_{\mathbb{R}^d}\rho \,dx=1$ and define $\rho_\...

**13**

votes

**1**answer

458 views

### Moduli space of linear partial differential equations

Is there a way to view "the space of all possible linear PDE's" as an algebraic variety with singularities?
This is in connection with a quote from someone on the web that I saw a long time ago. At ...

**7**

votes

**1**answer

161 views

### Does harmonic map heat flow of a curve always fully converge to a geodesic?

Consider a smooth closed curve $u_0$ in a compact Riemannian manifold $(M,g)$. Let $u_0$ evolve by harmonic map heat flow, $\partial_tu=\nabla_{\partial_su}\partial_su$, and call the result $u(t)$.
...

**2**

votes

**0**answers

72 views

### Analytic continuation of an NLS soliton

The attractive nonlinear Schroedinger equation $i \partial_t \psi = - \frac {1} {2} \Delta \psi - |\psi|^{p-1} \psi$ in the $H^1$-subcritical case $1 < p$, $\frac {d} {2} + \frac {2} {p-1} < 1$ ...

**2**

votes

**0**answers

62 views

### The Poisson equation

I see the following theorem in Lihe Wang's A geometric approach to the Calderon--Zygmund estimates
$$\triangle u=f,in \> B_2 \>(1)$$
Lemma 7: There is a constant $N_1$ so that for any $ε > ...

**2**

votes

**1**answer

87 views

### Removable set for Sobolev space

It is well known that if $\Omega\subset\mathbb{R}^{N}$ open, $F\subset\Omega$ closed, such that $\mathcal{H}^{N−1}(F)=0$,where $\mathcal{H}^{N−1}$ denotes (N-1) dimensional Hausdorff measure, then $W^{...

**5**

votes

**3**answers

193 views

### about the Hausdorff dimension of Removable singularities of PDE

There are some interesting phenomenons about removable singularities (or extension problems).
In the theory of functions of several complex variables, we know the classical Hartogs theorem:
Let f ...

**0**

votes

**0**answers

67 views

### Green functions for circular sectors

I would like to solve the Dirichlet boundary value problem
$$ (\Delta-k^2)u=0 \ \ \ \text{in $\Omega$} \\ u=f \ \ \ \text{on $\partial \Omega$} $$
where $\Omega$ is an infinite circular sector of ...

**4**

votes

**2**answers

215 views

### Question on PDEs which are related to certain geometric problems (e.g. Calabi conjecture, Gauduchon conjecture)

There are interesting symmetric functions $P_k$ arising from differential geometry and PDEs,
where $P_k$ is given by
\begin{equation}
\begin{aligned}
P_k(\lambda) = \prod_{1\leq i_{1}<\cdots < ...

**2**

votes

**2**answers

188 views

### Non-linear Basis for PDE's

Asked this on stack exchange and got no response, so I'll try here.
An idea popped into my head awhile ago while doing a project on non-linear effects of systems of coupled oscillators, but I'm not ...

**4**

votes

**0**answers

96 views

### An embedding question: Morrey spaces

Question. If $u\in L^1$ and $Du$ is in the dual of the Holder space $C^\alpha$, then is it possible to say $u$ belongs to some Morrey space $L^{1, \delta}$?

**0**

votes

**1**answer

42 views

### Strict positive type function on hypersurface also of positive type in neighborhood?

Let $u\in C^\infty(\mathbb{R}^n\times\mathbb{R}^n)$ be symmetric and of strictly positive type on some hypersurface $S \subset \mathbb{R}^n$ diffeomorphic to $\{0\}\times\mathbb{R}^{n-1}$. This means ...

**0**

votes

**0**answers

24 views

### Eigenfunction of Distorted Laplacian on Smooth compact domain

Setup
Suppose that $D$ is a compact star-shaped domain in $\mathbb{R}^d$ which is diffeomorphic to a closed $d$-dimensional ball in $\mathbb{R}^d$. Let $a(t,x)>0$ be a class $\mathscr{C}^{\infty}(\...

**1**

vote

**0**answers

47 views

### Solving Oblique Boundary value problem using Neumann Boundary Condition

\begin{align*}
&\frac{1}{2}a_{ij}(x)\frac{\partial^{2}u }{\partial x_{i}\partial x_{j}}+ b_{i}(x)\frac{\partial u}{\partial x_{i}}+ c(x)u=\lambda u ~~~in ~~\Omega\\
&\frac{\partial u}{\partial ...

**0**

votes

**0**answers

65 views

### Understanding the proof that $\Delta u = f(u)$ has a unique critical point on a convex domain

I am struggling to understand step 2 of the proof of Theorem 1 in [1]. It seems that the proof that the critical point is unique relies only on the fact that the nodal curves $$N_\theta = \{x\in\Omega\...

**3**

votes

**1**answer

133 views

### Universal decay rate of the Fisher information along the heat flow

I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation
$$
\partial_t u=\Delta u
$$
...

**1**

vote

**0**answers

65 views

### relative compact on nonlinear term

On the paper: Decay of Solutions to Nonlinear Schrodinger
Equations.
Let $u$ be a solution of the equation
$$Hu+|u|^2u=0,$$
where $H$ is a Schrodinger operator, i.e. $-\Delta+V$ and $V$ is a (...

**4**

votes

**0**answers

100 views

### Spectral gap for the Brownian motion with drift on a compact manifold

Let $M$ be a compact Riemannian manifold without boundary, $X$ a smooth vector field on $M$. Consider the Brownian motion $t\mapsto B_t$ on $M$ with drift $X$, so that its generator is $L=\Delta +X$. ...

**6**

votes

**1**answer

197 views

### is signed distance function real analytic for real analytic domains

If $\Omega$ is a real analytic domain in $\mathbb R^n$, is the signed distance function, $f$, defined by
\begin{equation}
f(x)=\begin{cases}d(x,\partial \Omega )&{\mbox{ if }}x\in \Omega \\-d(x,\...

**5**

votes

**1**answer

103 views

### Is the evaluation map from harmonic forms on the torus surjective on flat neighbourhoods?

In a nutshell:
Given a metric on the torus $\mathbb{T}^n$, can we extend any element $\sigma \in \bigwedge^k T_p^*\mathbb{T}^n$ to a global harmonic form?
Let $\mathbb{T}^n$ be the $n$-Torus. Fix ...

**0**

votes

**0**answers

55 views

### spherical mean and fractional laplacian

Define the spherical mean
$$v(r)= \frac{1}{ |\partial B_r(0) | } \int_{\partial B_r(0)} u \, dS$$
where $dS$ denotes integration with respect to spherical measure.
Is it possible to find the ...

**0**

votes

**0**answers

50 views

### Convergence in integral

I tried to prove that:
(f1) $f(0)=0$, $f\in C^1(0,\infty)$, $f'(s)>0$ for $s>0$, and there is a $p\in(2, 2N/(N-4))$ such that $\lim_{s\to \infty}f(s)/s^{p-2}<\infty$,
(f2) $F(s)=\int_0^sf(t)...

**0**

votes

**0**answers

27 views

### Existence of solutions to NLS: Local existence and boundedness

I was wondering when the following argument is valid:
Consider a nonlinear Schrödinger equation
$$i \partial_t \varphi = -\Delta \varphi+ N(\varphi)$$
where $N$ is a nonlinearity.
Often it is ...

**0**

votes

**0**answers

27 views

### trace embeddings and Sobolev inequality

I am reading the paper https://arxiv.org/pdf/0905.1257.pdf. Lemma 2.4. Equation (2.9) states that;
When $n=1$, for any $U \in H^{1}_{0, L} (\mathcal C);$ its trace $U(x, 0)$ is continuous embedded in ...