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.
4,467 questions
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Regularity structures-paracontrolled distributions: do they always work for sub-critical SPDE?
Stochastic PDE could be solved using either regularity structures or paracontrolled distributions, as long it's sub-critical.
I was wondering if this was proven, that is every sub-critical SPDE could ...
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Regularity structures vs Renormalization
What are the substantial differences in the theory of "Regularity Structures" versus perturbative renormalization from Quantum Field Theory?
The idea that to treat divergences inherent to ...
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Failure of regularity up to the boundary for a linear elliptic PDE
I asked a question before where I wanted a simple example where regularity up to the boundary fails for a linear elliptic PDE. I was presented an example with $\Omega = B(0,1) \backslash \{0\}$ (ball ...
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Question on ODE involving mollifiers from Taylor's book on PDEs
In Taylor's third book on PDEs chapter 16, the author discusses quasilinear symmetric hyperbolic systems of the form
$$\partial_{t}u=A^{k}(t,x,u)\partial_{k}u+g(t,x,u)$$
with some initial condition $u(...
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Examples showing Rellich-Kondrakov theorem fails for domains with non-Lipschitz boundary?
The Rellich-Kondrakov theorem requires the domain to have Lipschitz boundary, but after searching in many places, I have not found an example to show that this is necessary.
Can anyone share an ...
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Sufficient condition for interpolation
If we have a couple of two compatible banach spaces (in this sense) $(X,Y)$ and a sequence of Banach spaces $\{Z\}_{\theta\in[0,1]}$ which are intermediate between $X$ and $Y$ satisfying:
$Z_0=X$, $...
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Rate of convergence of mollified distributions in Besov spaces with negative regularity
Given a standard mollifier $\rho_\delta$ and a distribution $ u \in B^\alpha_{ p, p}$ with $\alpha<0$, $p \in [1, \infty]$ and $B^\alpha_{p,p}$ is a not-homogeneous Besov space, I'm trying to prove ...
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Integrability of modified diagonalizable Jacobian
I have a smooth function $f$ from $\mathbf{R}^N$ to $\mathbf{R}^N$. For each $x\in \mathbf{R}^N$ the Jacobian of $f$, $J_f$, is diagonalizable as
$$
J_f(x)=S(x)\Lambda(x) {S(x)}^{-1},
$$
where the ...
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How can i show that the last equality in the text is true?
Suppose that $v$ is critical point of
$$
f(u)=\frac{1}{2} \int_D|\nabla u|^2-\frac{1}{p+1} \int_D|u|^{p+1}, \quad u \in H_{0, \text{rad}}^1(D),\quad D(r, d)=\left\{z \in R^N: r^2<|z|^2<(r+d)^2\...
CommunityBot
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Estimating a solution to Euler-type ODE #2
This is a similar question to this but with a different ODE.
Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer, $R&...
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Finding thin plate spline subjected to boundary conditions
I am trying to formulate a problem as a PDE. What I want to know is if my formulation is correct, if it admits solution and what am I missing.
This question is related to : Thin-Plate-Spline ...
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Complex sum of squares of vector fields (hypoelliptic operators)
Consider a compact $M$ of dimension $n$. Consider real smooth vector fields $X_0, X_1, X_2,..., X_n$ on $M$ and consider the differential operator $\mathcal{L}_1 = \sum_{i = 1}^n X_i^2 + X_0.$
Now, by ...
CommunityBot
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Harmonic functions and monotonic decay
I have a general question surrounding certain harmonic functions.
I was able to solve the Laplace equation $\Delta f = 0$ in $\mathbb{R^3}$, subject to two spherical (equal radii) boundary conditions, ...
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de Giorgi iteration for the $p$-Laplacian as $p \to \infty$
I am trying to follow the proof of the main theorem of Part III of "Limits as $p \to \infty$ of $\Delta_p u_p = f$ and related extremal problems" by Bhattacharya, DiBenedetto, and Manfredi. ...
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Two-sided estimates of fundamental solutions of second-order parabolic equations
I am reading the paper Two-sided estimates of fundamental solutions of second-order parabolic equations, and some applications by F.O. Porper and S.D. Eidel'man. Below, the cited paper is
[2] :S.D. ...
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Heat conduction type equation in 4D
[I asked a similar question, Linear PDE, analytic continuation, Green's function and boundary conditions, and was told that a follow-up question should be a separate post.]
I'm interested in a ...
CommunityBot
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answer
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Estimating a solution to an Euler-type ODE
Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer and $a$ be a real number.
Let $u(r)$ be a function on $[1,\infty)$ ...
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Does local gradient collinearity imply factorization
Assume that we have two analytic functions $f(x,y)$ and $g(x,y)$ defined near $(0,0)$ on the plane. Let us also assume that $\nabla f =k \nabla g$, where $k=k(x,y)$ is also analytic and $\nabla f(0,0)\...
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Perturbation methods for stochastic/partial differential equations
I'm asking for a good reference on perturbation methods for stochastic and/or partial differential equations.
Something like this: Perturbation of a stochastic differential equation
I'm familiar with ...
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Regularity of solutions to an elliptic boundary value problem
Let $M = [1,\infty)\times S^2$. For an integer $k \geq 2$ and number $\tau<0$, define the space $L^2_{\tau}([1,\infty);H^k(S^2))$ to be all $H^k(S^2)$-valued functions $u$ on $[1,\infty)$ with $\...
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A fourth-order linear PDE
I am interested in the following type of $4$-th order linear PDE with $2$ variables (i.e., $x$ and $t$):
$$x^3 f_{xxxt}+ f =0$$
Does anyone know if this type of PDE already appeared in the literature? ...
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Generalized Fuchsian-type PDE
Consider
$$
\big(1+ t\partial_t\big) \left(\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x\right)A(x,t)+ {t\over (1-x)^3} A(x,t)=0
$$
with the initial condition $A(x,0)=1$. In a small $t$...
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Method of characteristics for higher order PDEs in more than two variables
I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
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Eigenvalues and eigenfunctions of the Laplace operator on entire plane
According to the answers in the the following questions: How to prove the spectrum of the Laplace operator? and What is spectrum for Laplacian in $\mathbb{R}^n$ , the spectrum of the Laplace operator $...
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Question about Neumann eigenvalues on manifolds
Question:
Let $\Omega\subset \mathbb{S}^2_+$ be any geodesically convex subset of the hemisphere $\mathbb{S}^2_+$. Then is it true that $\mu_1(\Omega)\geq \mu_1(\mathbb{S}^2_+)$ where $\mu_1$ is the ...
CommunityBot
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Estimating a potential
Let $ \Omega$ denote a smooth bounded domain in $R^N$ where $N \ge 3$ and we let $u \in C^3( \overline{\Omega})$. Let $ \delta(x) = \operatorname{dist}(x, \partial \Omega)$. For $ x \in \Omega$ (but ...
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Better regularity than $C^{1,\alpha}$ for $p$-harmonic maps into certain target manifolds
This is a follow up question I had while reading through this question and "Representing Homotopy Groups and Spaces of Maps by $p$-harmonic maps" by Shihshu Walter Wei.
We know that ...
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Definitions of weak solutions for quasilinear wave equations
I am learning the shock problem for the balance system (perhaps not conserved, see, e.g., "Ingo Muller, Tommaso Ruggeri. Rational Extended Thermodynamics") and just have a question on the ...
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Well-posedness of PDE with $\partial_{tt}\Delta u$ - like term
I am looking for direct hints or references for the establishment of existence of suitable weak solutions admitted by a class of problems of the following type: We search $u$ satisfying
$$
\begin{...
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3
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Existence of an extension operator $E: W_0^{s,p}(\Omega)\rightarrow W^{s,p}(\mathbb{R}^d)$?
If $\Omega$ is an open Lipschitz domain with bounded boundary then their exists a continuous operator $E: W^{s,p}(\Omega) \rightarrow W^{s,p}(\mathbb{R}^d)$ where $s\in (0,1)$ such that $Eu=u$ on $\...
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Galerkin scheme in $H^s_0(G)\subset L^2(G)\subset H^{-s}(G)$ ($s>0$)
What basis functions are usually choosen if one attempts to conduct a Galerkin finite element method given an evolution triplet $H^s_0(G)\subset L^2(G)\subset H^{-s}(G)$. Where $G$ is a sufficiently ...
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On an estimate in the paper by Donnelly and Fefferman
I was reading the following paper by Donnelly and Fefferman https://link.springer.com/content/pdf/10.1007/BF01393691.pdf which essentially deals with the Hausdorff dimension bound of the nodal sets ...
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Image of trace operator on $W^{2,1}(\mathbb{R}^2)$
It is known that for a domain $\Omega\subset \mathbb{R}^2$ with $C^1$ boundary $\partial\Omega$, that the trace operator is bounded and surjective from $T:W^{1,1}(\Omega)\to L^1(\partial\Omega)$.
For ...
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On the weak derivative of $|u|^{(p-2)/2}u$
Let $u$ be a function such that $|u|^{(p-2)/2}u$ is in $H^1_0(G)$, $G$ is open and $p>2$.
How can I show that
$$
D(|u|^{(p-2)/2}u)=p/2|u|^{(p-2)/2}D(u) \label{1}\tag{1}
$$
or how can I show that, ...
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Derive elliptic maximum principle from weak derivatives
Let $U$ is a connected open set, and $a^{ij}, c^i \in L^\infty (U).$ $a^{ij}$ satisfies the uniform ellipticity condition. Suppose that $u\in H^1(U) \cap C(\overline U)$ satisfies the condition that
$$...
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A question about the eigenfunction method and the notion of solution - distributional solution
I have a question about how a passage was made in the calculation of passage (2.5) in the calculation below. To introduce context, the author in the paper (full work) is trying to demonstrate that ...
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Systems of (hyperbolic) 2nd order PDEs with lower order constraints
Certain surfaces in mechanics are endowed with the fundamental forms
\begin{align}
\text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\
\text{II} &= \alpha\left(\...
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1
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A bilinear estimate with a simple one-dimensional oscillatory integral kernel
Let $f\in L^{p}(\mathbb{R})$, $1\leq p\leq 2$.
I am trying to show that
$$\int_{\mathbb{R}}\int_{\mathbb{R}}
\,K(y,z)\,
\frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,...
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Is $L^2([a,b]; L^2(S^2))$ the same as $L^2([a,b] \times S^2)$?
The space $L^2([a,b];L^2(S^2))$ is a Banach space with respect to the norm
$$\left\Vert f \right\Vert_1^2 = \int_{a}^b \left\Vert f(r) \right\Vert_{L^2(S^2)}^2 dr$$
The space $L^2([a,b]\times S^2)$ ...
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Why are viscosity solutions useful solutions?
I refer to definition of viscosity solution in user's guide to viscosity solutions of second order partial differential equations by Michael G. Crandall, Hitoshi Ishii and Pierre-Louis Lions.
...
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Feynman–Kac formula for other operators
I recently came across the Feynman-Kac formula which states that given an open domain $\Omega\in\mathbb{R}^n$ and $f \in L^2(\Omega)$
where $x \in \Omega$ and $t > 0$, then
$e^{t\Delta_D}f(x) = ...
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Why does failure of boundedness of this operator for $p<q$ implies its failure for $p>q^{\prime}$?
I am reading the paper "P.Sjolin, Convolution with Oscillating Kernels, Indiana University Mathematics Journal Vol. 30, No. 1 (1981), pp. 47-55" where $L^p-L^p$ boundedness of the operator
$...
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Techniques to estimate PDE which are elliptic in some directions and degenerate in others
I am interested in a family of PDE which have defeated my (admittedly rather naive) attempts to prove any regularity or stability estimates. These are systems of PDE which are elliptic in some ...
- 708
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0
answers
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Formulation of multipoint constraints using Lagrange multipliers for a time dependent problem (with the Finite Element Method)
Intro
Suppose we have the following static linear equations (e.g. of an elastostatic problem):
$$\mathbf{K}\boldsymbol{u}=\boldsymbol{f}$$
We want a multipoint constraint of the type
$$\boldsymbol{\...
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1
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How is this interpolating curve well-defined in the minimizing movement scheme?
Let $\Omega$ be a compact domain of $\mathbb R^d$. Let $\mathcal P (\Omega)$ be the space of probability measures on $\Omega$. For each $\tau >0$, let $(\varrho^\tau_{(k)})_{k \in \mathbb N} \...
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Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces
It is known [1, proposition 3.32] and a classical trick in PDEs that, in any uniformly convex Banach space $X$, weak convergence $x_n\rightharpoonup x$ together with convergence of the norm $\|x_n\|_X\...
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1
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Classification of homogeneous distributions
On page 92 of these notes, there is a discussion on how to find the fundamental solution to the D'Alembertian operator. It is firstly proposed that $c_n(t^2 - |x|^2 )^{-(n-1)/2}$ may be a good ...
- 115
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0
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Understanding the effect of PDE solution on critical strip?
I would like to understand a little bit about how to interpret and construct $1$-parameter gamma factors that are dynamical - that is they are particular solutions to linear PDE's. Some possible ...
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0
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Trouble understanding Lax method for KDV equation for inverse scattering method
I am trying to learn the Lax pair condition on my own so that I can eventually learn the inverse scattering method. I am following a paper by Tuncay Aktosun ("Inverse scattering transform and the ...
4
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2
answers
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Hörmander's hypoellipticity theorem for complex coefficients
Hörmander's theorem says that if $L = \sum _{i=1} ^r X_i ^2+ X_0 + f$ on some open subset $U \subseteq \Bbb R$ has the property that the Lie algebra generated by $\{X_0, \dots, X_r\}$ at every point ...
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