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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
1answer
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
0answers
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
0answers
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
0answers
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
2answers
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
0answers
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
0answers
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
1answer
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
2answers
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
3answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
3answers
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
0answers
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
2answers
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
2answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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 ...