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Questions tagged [ap.analysis-of-pdes]

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

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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 ...
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59 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 ...
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45 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{{\...
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2answers
247 views

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 $$-\...
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0answers
155 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(...
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58 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? ...
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52 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 ...
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1answer
139 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 ...
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2answers
214 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 ...
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0answers
72 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 ...
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42 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(\...
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1answer
137 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 ...
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2answers
237 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 ...
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3answers
203 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 ...
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0answers
46 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 ...
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0answers
69 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-...
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114 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 $$\...
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0answers
40 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 ...
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0answers
79 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_\...
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1answer
494 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 ...
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1answer
188 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 $u(t)$. ...
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0answers
89 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 $1 < p$, $\frac {d} {2} + \frac {2} {p-1} < 1$ ...
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1answer
176 views

The Poisson equation

I see the following theorem in Lihe Wang's A geometric approach to the Calderon--Zygmund estimates $$ \triangle u=f\quad in \quad \> B_2. \>\quad \quad \quad \quad (1) $$ Lemma 7: There is a ...
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1answer
108 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^{...
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3answers
220 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 ...
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0answers
72 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 ...
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2answers
227 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 < ...
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3answers
282 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 ...
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0answers
104 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 $L^{1, \delta}$?
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1answer
45 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 ...
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25 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}(\...
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0answers
56 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 ...
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77 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\...
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1answer
138 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 $$ \partial_t u=\Delta u $$ ...
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0answers
68 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 (...
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0answers
112 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 $L=\Delta +X$. ...
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1answer
202 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,\...
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1answer
116 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 ...
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0answers
57 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 ...
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0answers
32 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 ...
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0answers
30 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 ...
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1answer
153 views

improvement of flatness in the regularity of minimal surfaces

Recently,I am reading Savin's celebrated theorem about improvement of flatness in proving the regularity of minimal surfaces. I have some questions. 1.How to show the boundary of a minimal set ...
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4answers
180 views

PDE with Laplacian and squared of the gradient

Let $u$ be a real function in $\mathbb{R}^2$. Does anybody know that the following PDE $$\Delta u+|\nabla u|^2=0$$ has any non-constant general solution or not? It would be appreciated if any one ...
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1answer
45 views

Existence of bisolutions for wave operators on globally hyperbolic Lorentzian manifolds

Let $(M,g)$ be a globally hyperbolic Lorentzian manifold with Lorentzian volume density $dV$ and $\Sigma$ a Cauchy hypersurface, i.e. each inextendible causal curve hits $\Sigma$ exactly once. ...
2
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1answer
140 views

Evans-Krylov theorem

Do there exist estimates for nonconcave functionals similar to Evans-Krylov theorem in chapter 6 of Fully nonlinear elliptic equations by Luis A.C affarelli and Cabre? Perhaps there is a ...
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0answers
19 views

Subadditivity on nonlocal minimum problem

Recently, I read a paper about the concentration compactness. I want to general it that add a global term. For example: Let $$ E(u)= ||\nabla u||_{L^2(\mathbb R^n)}^2 \int _{\mathbb R^n} |\nabla u|^...
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0answers
50 views

Uniqueness of solution of linear PDE of first order

Let $\vec u\colon \mathbb{R}^n\times \mathbb{R}\to \mathbb{R}^k$ be at least $C^1$- smooth vector valued function. Assume they satisfy a first order linear equation $$\partial_t \vec u(x,t)=\sum_{j=1}^...
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1answer
162 views

PDE’s whose solutions can be presented using path integrals

It is well known that solutions of the Schroedinger equation and of the heat equation can be presented using path integrals: $$\psi(x,t)=\int K(x,t;y,0)\psi(y,0)dy,$$ where the kernel $K(x,t;y,0)$ is ...
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2answers
443 views

Has it been proved that weak solutions to the Navier-Stokes equations are non-unique, and does this prove that the Navier-Stokes are not valid?

This preprint claims that, for finite kinetic energy initial solutions, uniqueness of weak solutions to the Navier-Stokes equations doesn't hold: https://arxiv.org/abs/1709.10033 What's the current ...
4
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0answers
70 views

Anderson Localization and Homogenization theory

I originally asked this on Mathematics Stack Exchange but realized it might be better to ask it here. The question is mostly related to homogenization theory in mathematical physics. $\textbf{...