0
votes
1answer
178 views

Theorem with an example [on hold]

i have this theorem in the paper they gives an example: but here $H_1$ is not satisfied ! How to correct it please?
3
votes
2answers
114 views

Existence for ODE in Banach space (accretive operators and Crandall-Liggett)

There is a theory of mild solutions $u \in C^0(0,T;X)$ where $X$ is a Banach space for equations of the form $$\frac{du}{dt} + Au = f$$ where $A$ is an accretive nonlinear operator under some ...
1
vote
0answers
110 views

Bound for a certain integral expression

I am working to establish an estimate in $X^{s,b}$ spaces to prove local well-posedness of a certain equation, and I need to consider some sub-cases. In particular, I wish to show that the following ...
1
vote
1answer
113 views

Pohozaev result for equations with weights

I am interested in nonnegative solutions of $-div( e^{-\gamma(x)} \nabla u(x)) = e^{-\gamma(x)} u(x)^p$ in $\Omega$ with $ u=0$ on $ \partial \Omega$. Or instead the equation $ -\Delta u + ...
2
votes
1answer
111 views

Is Poisson's kernel integrable?

Let $E$ be a smooth domain. Green's function is defined as $G(x,y)=F(x,y)-\Phi(x,y)$ where $F$ is the fundamental solution to the Laplace equation. For a fixed $x\in E$, $\Phi(x,\cdot)$ is a harmonic ...
1
vote
3answers
241 views

What are the basis functions for a product space?

Let $X=L^1\left([0,1]^3\right)$, for numerical purpose, what are the possible basis function for $X$? In finite element method, the basis functions are tooth functions, or polynomial functions. Is ...
2
votes
2answers
155 views

Regularity of a nonlinear ODE [Traveling wave solutions of parabolic systems]

In the book of Volpert on Traveling wave solutions of Parabolic Systems (AMS), one reads "the following assertion is readily proved and we shall not discuss it in detail". The same result is tacitely ...
4
votes
1answer
262 views

Pseudo-differential operators with compactly supported symbols

If the symbol $p(x,\xi)$ of a pseudodifferential operator $P$ has compact $x$-support, then for any Schwartz function $f$, $Pf$ has compact $x$-support. Is the reverse true? Namely that if some PDO ...
2
votes
0answers
146 views

A contradiction to do with continuity? (involves chain rule)

Suppose for each $t$, $S(t) \subset \mathbb{R}^n$ is a domain (hypersurface). We have a diffeomorphism $D^0_t:S(0) \to S(t)$ for each $t$ such that it solves the ODE $$\frac{d}{dt}D^0_t(\cdot) = ...
1
vote
2answers
95 views

Bound deg 3 partial differential operator on Laplace eigenfunction?

I am no expert on PDE and analysis but I am looking for certain technique from PDE. Let $D_2$ be the Laplace operator and $f$ is an eigenfunction, i.e., $D_2 f=\lambda f$ for some $\lambda>1$. (or ...
3
votes
0answers
112 views

A microlocal representation for quantum operator dynamics

In Maciej Zworski's book $\textit{Semiclassical Analysis}$, an important step in proving $L^p$ bounds on quasimodes is deriving a microlocal oscillatory integral representation formula for families of ...
2
votes
1answer
131 views

First integrals of a 3D incompressible flow

Let $\Omega$ be an unbounded periodic smooth domain of $\mathbb{R}^3$. We are Given an incompressible vector field $q:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$ (i.e. $\nabla\cdot q\equiv 0$ ...
3
votes
2answers
180 views

Elliptic Harnack inequality for 1D Schrodinger operator?

For a nonnegative polynomial $V: \mathbb{R} \to \mathbb{R}$, write $H = -\Delta + V$. I am wondering if there is an elliptic Harnack inequality for H. That is: There exist $C_{H} > 0$ and ...
0
votes
3answers
434 views

I have this linear PDE…

Hi, The PDE in question is: $A P_{yy}(y,z) + B P_{zz}(y,z) + ( [ C y -D z] P(y,z) )_y + ( [ D y + C z ] P(y,z) )_z=0,$ where subscript $y,z$ indicates derivatives and $A,B,C,D$ are real. The PDE is ...
1
vote
0answers
178 views

Solving a PDE involving a mixed derivative for a partial derivative

Consider a PDE of the form \begin{equation} \frac{\partial^2u}{\partial p\partial t}=F\left(\frac{\partial u}{\partial p},u,p\right) \end{equation} or \begin{equation} \frac{\partial^2u}{\partial ...
1
vote
1answer
145 views

weak*closure of {f:||f||=1} in dual.

What is the weak* closure of {f:||f||=1}? I am sure this set is not closed in weak* topology. So what is the weak* closure of this set. Thanks.
1
vote
2answers
295 views

Replacing large-dimensional ODE systems with one PDE [closed]

Is it possible to replace a large-dimensional system of differential equations with one partial differential equation?
1
vote
0answers
56 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, ...
6
votes
0answers
88 views

Lagrangean uniqueness versus Eulerian uniqueness

(1) Lagrangean description. Let us consider a $N\times N$ system of autonomous ODE: $$ \dot x=a(x),\quad \mathbb R\ni t\mapsto x(t)\in \mathbb R^N,\quad a:\mathbb R^N\rightarrow \mathbb R^N. $$ ...
2
votes
2answers
394 views

Easy question on Sobolev spaces

I understand that this question would be trivial for experts, sorry for that, I just need to clarify things. So let $S(\mathbb{R}^n)$ denote the Schwartz space on $\mathbb{R}^n$ and $W_p$, $W_q$ are ...
6
votes
0answers
251 views

Uniqueness for a non-local differential equation

Question:Fix $\epsilon>0$. Consider the differential equation, defined for functions $f(t,x)\in C^\infty([0,\epsilon]\times[0,\epsilon])$ defined by $$\frac{\partial}{\partial t} ...
4
votes
2answers
214 views

Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform

Question: Suppose $a(x,y)\in C^\infty([0,1]\times [0,1])$ and suppose $$\sup_{\lambda>1} \bigg|\lambda\int_0^1 e^{\lambda x} a(x,1/\lambda)dx\bigg|<\infty.$$ Is $a(x,0)=0$, $\forall x\in[0,1]$? ...
0
votes
1answer
129 views

What is the limit of the derivative of the harmonic extension/Dirichlet solution in $C^{1,\alpha}$ cases ?

Let $f:\mathbb{S}^1 \to \mathbb{S}^1$ be an oreintation-preserving homeomorphism. Denote by $H(f)$ the complex harmonic extension/solution in $\mathbb{D}$ to the Dirichlet problem with boundary data ...
1
vote
0answers
130 views

Weyl quantization and convexity

Let $C$ be a convex subset of $\mathbb R^{2n}$ and $\mathbf 1_C$ be the characteristic function of $C$. Is it true that $$\forall u\in\mathscr S(\mathbb R^n),\quad \langle\mathbf ...
3
votes
3answers
337 views

Continuity with values in L^2

Hi, let $T>0$, $\Omega\subset\mathrm{R}^n$ be a bounded smooth domain and suppose $$u\in L^2(0,T;W^{1,2}(\Omega))\cap L^\infty((0,T)\times\Omega))\ \text{and } \partial_tu\in ...
1
vote
2answers
246 views

how to solve a singular integral equation involving the kernel $1/x$

Dear all, Suppose we know that $f(x)$ is nonnegative and Hölder continuous at zero with exponents $1/2$. We also know that $$ f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0, $$ ...
1
vote
1answer
445 views

Solutions to Heat Equations with Obstacles!

Consider a closed Riemannian manifold $(M,g)$ and a positive function $\psi: M \to R$. Fix a point $p \in M$, I have been struggling to construct a solution to the heat equation, $\partial_t u = ...
23
votes
3answers
1k views

Which differential equations allow for a variational formulation?

Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation $$ ...
1
vote
1answer
136 views

Uniform equicontinuity of a family of indefinite integrals

Let $f_k$ be a sequence of measurable functions on $\mathbb{R}^k$ where $k > 1$. (Let us be generous and also assume that $f_k$ is locally integrable.) Does anyone know what the phrase uniform ...
12
votes
6answers
2k views

Square roots of the Laplace operator

In several places in the literature (e.g. this paper of Caffarelli and Silvestre), I've seen an integral formula for fractional Laplacians. I'd like to understand it. In this question, I'll stick to ...
1
vote
1answer
288 views

sobolev embedding theorem in the smooth metric measure space

we know the sobolev embedding theorem of Saloff-Coste $\Big(\int_B|F|^{2q}d\mu\Big)^{\frac1q}\le e^{C(1+\sqrt KR)}V^{-2/n}R^2\int_B\Big(|\nabla F|^2+R^{-2}F^2\Big)d\mu $ wtih $Ric\ge-(n-1)K$, for ...
2
votes
1answer
384 views

Distributional derivative is locally integrable - then the distribution as well?

Given a distribution $T \in D'(\mathbb{R})$ such that the distributional derivative $\partial T \in L^1_{loc}(\mathbb{R})$. Can one deduce that $T \in L^1_{loc}(\mathbb{R})$ as well? Or can anyone ...
7
votes
1answer
332 views

Mean value property with fixed radius

Let $f$ be a continuous function defined on $\mathbb{R^n}$. It is well known that both the spherical mean value property (MVP) of $f$, i.e. $$f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}f,\ ...
0
votes
0answers
83 views

A slightly subcritical elliptic equation on the ball; blow-up behavior near zero

I am interested in positive ground state solutions of the following elliptic pde: $-\Delta u(x) = u(x)^{p-\epsilon} $ in the unit ball $B$ in $ R^N$ with $ u=0$ on $ \partial B$. Here $ ...
2
votes
0answers
152 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 ...
1
vote
2answers
159 views

vector valued BVP for ODE's

I am dealing with a vector valued second order homogeneous BVP: $\ddot u(t) = A(t)\dot u(t) + B(t)u(t)$ with $u(0)=u(1)=0.$ where $A$ and $B$ are $n \times n$ matrices with smooth coefficients and ...
4
votes
1answer
586 views

Results about existence/uniqueness of solution to Euler-Lagrange equations?

While studying calculus of variations, there is one question that I feel is missing in the texts I'm reading: What can we say about the existence and/or uniqueness of solutions to Euler-Lagrange ...
3
votes
1answer
333 views

Regarding Discrete Eigenvalues

For many eigenvalue problems for differential operators (for example the quantum harmonic oscillator (HO)), unless we impose some behaviour at infinity, the eigenvalues will not be discrete. But, ...
10
votes
1answer
379 views

Lie $2$-groups and differential equations

I was reading the abstract of a recent preprint (Division Algebras and Supersymmetry III by Juhn Huerta), and I wondered if something much simpler than what he was talking about had been worked on: ...
0
votes
1answer
755 views

Functionals continuous with respect to weak convergence

It's well known that a functional of the form $u \mapsto \int f(u) dx$ is continuous with respect to weak convergence (say weak* convergence in $L^\infty$) if and only if the function $f$ is affine. ...
3
votes
0answers
145 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
1answer
568 views

Endpoint Strichartz Estimates for the Schrödinger Equation

The non-endpoint Strichartz estimates for the (linear) Schrödinger equation: $$ \|e^{i t \Delta/2} u_0 \|_{L^q_t L^r_x(\mathbb{R}\times \mathbb{R}^d)} \lesssim \|u_0\|_{L^2_x(\mathbb{R}^d)} $$ $$ 2 ...
7
votes
1answer
393 views

What would the best treatment of Gehring's lemma look like?

In a course about elliptic regularity probably one sooner or later stubles into the reverse Holder inequalities, and has to introduce the Gehring lemma, which in one of its many versions improves a ...
8
votes
0answers
231 views

Finding a dimension-free bound for a certain multiplier on Euclidean space

The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...
1
vote
1answer
757 views

Sobolev Embedding Theorems

I am getting a little confused about the huge number of slight variations on the Sobolev Embedding Theorem. Let $\Omega\subseteq\mathbb{R}^n$ be a bounded Lipschitz domain and suppose that $f\in ...
2
votes
2answers
319 views

Seminorms in sharp Garding's inequality

When working with symbols of limited regularity, what is exactly the number of seminorms of the symbol $Re (a) \geq 0$ that one needs in the sharp Garding's inequality: $Re \langle a(x, D) u, u ...
3
votes
3answers
366 views

Maximum principle corner

Hello, consider a parabolic boundary value problem, for instance $-\partial_tu+\Delta u=0$, in $\Omega$, $\partial_\nu u=0$ on $\partial\Omega$, in a domain $Q=(0,T)\times\Omega$, where ...
0
votes
1answer
292 views

convergence of metrics

Hi, I have the following question: take a Riemannian manifold M, with a family of smooth metrics $g(t)$ in $[0,T)$, call $D_0$ the Levi-Civita connection of $g(0)$ and assume that for every $m\geq 0$ ...
2
votes
2answers
200 views

Poisson equation in the plane

Hello, as I'm not an analyst, I'm having difficulties with the following, certainly well-known problem: one is given the PDE $\Delta u(x,y)=\sqrt{x^2+y^2}$ in the "region" $x^2+y^2\leq1$ with the ...
3
votes
0answers
328 views

problem with non linear pde

I have the following pde which i cannot solve. any suggestions, tips on how to approach a solution? $$\left(1-x^3 \frac{\partial y}{\partial x} \partial_{y}\right) f(y(x))-1/4 \left(1-\frac{1}{x^3 ...