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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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1answer
109 views

Existence of unique critical points to second order elliptic PDEs

Let $\Omega\subset\mathbb{R}^2$ is bounded and convex and $\partial\Omega$ be smooth. Consider the second order elliptic PDE (1) $$ \begin{cases} Lu = f &\text{ on } \Omega\,,\\ u=0 &\text{ ...
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0answers
122 views

Compact embedding result

Let $\tau$ and $\ell$ be positive numbers. We know that the space $H^2(0,\ell)\cap H^1_0(0,\ell)$ is compactly embedded into $L^6(0,\ell)$. Now, is the space $L^2(0,\tau;H^2(0,\ell)\cap H^1_0(0,\ell))$...
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1answer
81 views

A solution to the Navier-Stokes equation that is defined for on $[0,T]$ with $T$ large is global?

Let $u_0 \in \dot{H}^{1/2}(\mathbb{R}^3)$. The Fujita-Kato theorem gives rise to a local unique solution $(t,x) \mapsto u(t,x)$ to the Navier-Stokes equations $$\left\{ \begin{array}{ccc} \partial _t ...
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0answers
131 views

Simple proof for Fefferman-Phong inequality?

The following inequality was proved firstly by Fefferman in the paper: The uncertainty principle, Bulletin of the AMS, 1983. Then it was improved by several authors. The proofs presented in these ...
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0answers
83 views

How do we know the mollification is in the Sobolev space?

The first theorem of section 5.3. in Evan's PDE discusses approximating a function in $W^{p, k}$ by it's mollifications. Suppose $k$ is a positive integer, $1\leq p <\infty$ and $U$ is an open ...
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1answer
112 views

Schrödinger operator with Coulomb potential

The free Laplacian $-\Delta$ has absolutely continuous spectrum $[0,\infty).$ The Coulomb Hamiltonian $H=-\Delta-\frac{1}{\vert x\vert}$ on $L^2(\mathbb R^3)$ has absolutely continuous spectrum $[0,\...
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47 views

Elliptic Dirichlet BVP's for regions with multiple boundary components

Apologies for the vague title, it was getting rather long so I decided to just explain more in the body of the text. I am curious about the state of understanding for existence and uniqueness of ...
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0answers
22 views

soliton solution of nonlinear convection-diffusion-reaction equation system?

I am interested in a flame propagation in 1D which is governed by a nonlinear convection-diffusion-reaction equation system: $$ u_t+C(u)u_x+(D(u)\cdot u_x)_x + R(u) = 0,x\in (-\infty,+\infty) \\ u(-\...
4
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1answer
125 views

Method of characteristics beyond the Lipschitz setting

I have come across the following easy-looking problem that is driving me mad. I have a family of measures (on the real line $\mathbb R$) $\{\mu_t\}_{t>0}$ which is uniformly bounded (the measures ...
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1answer
124 views

Elliptic regularity of harmonic forms in $L^1$

$\newcommand{\M}{M}$ This is a cross-post. I am looking for a reference for the regularity of harmonic forms which belong to $L^1(M)$. Explicitly, let $\M$ be a smooth oriented Riemannian manifold. ...
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83 views

What is a relation between energy space and $L^p_s-$Sobolev spaces?

We define energy space $$E= \left\{ f\in \mathcal{S}'(\mathbb R^d): \|\nabla f\|_{L^2} + \|xf\|_{L^2} < \infty \right\}.$$ and Sobolev spaces $$L^p_s(\mathbb R^d)=\{f\in \mathcal{S}'(\mathbb R^...
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73 views

Is every “higher-order” harmonic morphism conformal?

$\newcommand{\TM}{\operatorname{TM}}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\TM}{\operatorname{T\M}}$ $\newcommand{\TN}{\operatorname{T\N}}$ $\newcommand{\TstarM}{...
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0answers
41 views

Distributions and interchange of integrals under a PDE

For some $\eta>\epsilon>0$ (small), I have a double (actually many-dimensional) integral of the form $$ F(x) = F(f;x) = \int_{\Bbb{R}} \int_{\Bbb{R}} f(y) h(x,y,z) dy dz, \qquad x>0$$ where $...
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1answer
82 views

Quick question on the constants involved in heat kernel upper bounds

Let $M$ be a compact Riemannian manifold without boundary, and let $\Delta$ be the Laplace-Beltrami operator on $M$. It is known that for small $t$, let's say, $0< t < t_0(M, g)$, the heat ...
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4answers
723 views

History of ODE and PDE reference request

Is there any reference (book or articles) which made the history (up to the modern times) and the conceptual development of Ordinary Differential Equations and Partial Differential Equations? It will ...
7
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1answer
102 views

Estimate for radius of convergence of solutions given by Cauchy-Kovalevskaya Theorem

I'm sure you can extract it from the proof, but does anyone know of a reference where the radius of convergence (in terms of radius of convergence of the initial data and PDE) of the solution given by ...
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0answers
115 views

Different types of a test functions in weak solutions of a PDEs

In the literature about PDEs it is easy to find books that talk about weak solutions of a partial differential equations. A short reminder of the most usual definition is given bellow. Let's ...
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1answer
109 views

Closure of tensor product /tensor product semigroup

In this reference the following claim is made in Remark 2 Let $A,B$ be closable operators on Banach spaces $X,Y$, then $A \otimes 1$ and $1 \otimes B$ are closable operators on the Banach space $X \...
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1answer
176 views

On the Calabi-Yau conjecture for minimal surfaces

Colding and Minicozzi proved that any embedded minimal surface in $\mathbb{R}^3$ with finite topology must be proper and thus it can not be bounded. Is it possible to remove the assumption "finite ...
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0answers
879 views

A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it

Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
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0answers
167 views

Sobolev space under Mellin transform

The Mellin transform is known to be an isomorphism see wikipedia between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$ where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...
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2answers
200 views

A basic stability question

Let $u_i \in C^1(\Omega)$ with $|\nabla u_i|>0$ in a simply connected region $\Omega$ with connected boundary, and $u_1=u_2$ on $\partial \Omega$. Assume $$ \nabla u_i(x) \cdot V_i (x)=|\nabla u_i(...
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0answers
87 views

inverse of sobolev riemannian metric still sobolev?

Given a covariant riemannian metric of certain sobolev class (i.e. with square-integrable weak derivatives up to a large enough integer order) on a (compact, if necessary) finite-dimensional smooth ...
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2answers
237 views

Can we specify the value of harmonic forms at a point?

Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $1 < k < d$ be fixed. Let $p \in M$, and let $\alpha_p \in \bigwedge^k(T_pM)^*$. Does there exist an open ...
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3answers
495 views

A conformal map whose Jacobian vanishes at a point is constant?

Let $f:M \to N$ be a smooth weakly conformal map between connected $d$-dimensional Riemannian manifolds, i.e. $f$ satisfies $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}_{TM}$. Assume $d \ge 3$ ...
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2answers
295 views

Hölder continuity for operators

Let $x,y$ be positive real numbers then $$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$ we obtain $1/...
4
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0answers
146 views

gaussian upper bound on spherical heat kernel

It is known that the heat kernel on n-sphere satisfies $p_t(x,y)\leq Ct^{-n/2}e^{-d(x,y)^2/5t}$ for all $t\in (0,T)$. Can something be said about how big C needs to be?
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2answers
163 views

Improving equi-integrability for a family $\mathcal F$ in $L^1(\Omega)$

Let $\mathcal F$ be a weakly compact subset of $L^1(\Omega)$. Dunford–Pettis theorem says that $\mathcal F$ is uniformly integrable. Also, by de la Vallée-Poussin theorem we can find an increasing ...
4
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1answer
176 views

Wavefront set and Duhamel's principle

Consider the Cauchy problem: $$ \frac{\partial u}{\partial t} + \mathrm{i}\mkern1mu A(x,D_x) u = f \quad 0< t < T; \qquad u = u_0 \quad \text{when}\; t = 0, $$ where $A$ has real principal ...
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0answers
56 views

Constant in trace theorem for balls

Consider the standard open ball $B_r:=\left\{x ; \left\lvert x \right\rvert \le R \right\}.$ The trace theorem tells us any function in $W^{k,p}(B_r)$ can be restricted to a function $W^{k-1,p}(\...
7
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1answer
227 views

Proof of Green's formula for rectifiable Jordan curves

$\newcommand{\Ga}{\Gamma}$ I am trying to find a proof of Green's formula for rectifiable Jordan curves $\Ga$ (and the corresponding interior regions $R$). There is a proof by Ridder, followed by ...
3
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0answers
62 views

Is a relatively weakly compact subset of $W^{1,1}(\Omega)$ metrizable?

Let $\Omega$ be a domain with smooth boundary. Let $S\subset W^{1,1}(\Omega)$ be a relatively weakly compact set. Is it true that $(S,w)$ is metrizable? Since $S$ is relatively weakly compact, it ...
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1answer
390 views

Nonlinear Schrödinger equation with discrete Laplacian

In the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$" by Colliander, Keel, Staffilani, Takaoka and Tao it is argued in the beginning ...
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0answers
64 views

Sobolev embeddings for vector-valued functions

I would like to know if there is a simple extension of the standard Sobolev embeddings for functions taking values in another Euclidean space. In particular, let $\Omega \subset \mathbb{R}^n$ be a ...
2
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1answer
155 views

Reference request for weak solutions of an Elliptic PDE

Edit : I just learned that all weak solutions are $C^\infty$, so this question, by Willie, seems more appropriate than the current one. I want to find weak, non trivial, continuous, solutions of $$\...
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0answers
124 views

Support of a microlocal defect measure

I'm trying to complete the proof of the Theorem 6.1 in the notes https://www.math.u-psud.fr/~nb/articles/coursX.pdf, which ensures, under certain conditions, that the support of the microlocal defect ...
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1answer
151 views

Applications of the Calderon-Zygmund theory to PDE's

I am planning to build a PDE topics course focussing on the Calderon-Zygmund theory. I know some important applications of the Calderon-Zygmund theory to elliptic PDEs, but I don't know enough to get ...
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0answers
76 views

For what functions does Nash inequality becomes equality?

For what functions does Nash inequality becomes an equality? Also any comment on the regularity of these functions (weak solutions to equality)? Also same question about Poincare inequality.
2
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1answer
146 views

*Full proof* references for Markov generators with various boundary conditions

(Note: I've migrated this question from math.stackexchange, as the lack of answers there made me believe it was perhaps too advanced for that forum.) Consider the one-dimensional heat equation $$\...
3
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3answers
217 views

Decay estimate for the heat equation: $\sup_{t>0}\int_{\mathbb{R}} t^\alpha |u_x|^2\ dx$

Let $u$ be a solution of the heat equation $$u_t - u_{xx} = 0, \quad t>0, x \in \mathbb{R}$$ with initial data $u(0,\cdot) = u_0$. Fix $\alpha >0$. How can I estimate (without using explicitly ...
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0answers
42 views

von Neumann stability vs eigenvalues of amplification matrix

My, limited, understanding of the stability analysis of PDEs is that broadly speaking there are two methods: von Neumann analysis which looks at the growth the error of the solution, as described here ...
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0answers
51 views

Solvability of Neumann boundary problems with singular boundary data $g \in (H^{1})^{*}$

I have a question on the solvability of Neumann boundary problems with singular data. To state my question, let $\Omega$ be a bounded Lipschitz domain (open and connected) in $\mathbb{R}^n$. In the ...
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0answers
181 views

Absence of fixed points

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function $$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \frac{xy}{(x^2+y^2+1)} \ dx$$ where $x_0$ is an arbitrary but fixed ...
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2answers
73 views

Dirichlet problem for capillary equation over convex domain

Let $\Omega \subseteq \mathbb{R}^2$ be a bounded convex domain with piecewise smooth boundary. Let $\phi :\partial \Omega \to \mathbb R$ be a continuous function. Let $L$ be a quasilinear elliptic ...
3
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1answer
249 views

Function square-integrable

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function $$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$ where $x_0$ is an ...
2
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3answers
267 views

Uniqueness of solution depending on constant?

I am a physicist and I am aware that this forum is for professional mathematical questions, but please be not too hard on my notation. I encountered the following integral equation for functions $f:[...
2
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1answer
150 views

Parabolic variational inequality: regularity of the time derivative in $L^2(0,T;H)$?

Let $V \subset H \subset V^*$ be a Gelfand triple of Hilbert spaces. Take $f,\psi \in L^2(0,T;H)$ and consider the VI: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that $$u(t) \leq \psi(t) ...
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1answer
69 views

Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)

I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D image of my testsetting. I want to verify and compare different Discretizations of the ...
1
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2answers
145 views

Method of characteristics for 2x2 systems

In the literature it is easy to find books that show, step by step, how to get the solution of partial differential equation using various techniques, but it is not so easy to do the same for PDE ...
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0answers
103 views

How can I obtain this inequality (from Evan's PDEs)

I am self studying PDEs from Evans' "Partial Differential Equations" textbook. Currently, I am going through Theorem 1 from Section 5.7 (Rellich-Kondrachov compactness theorem) and am having ...