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.
4,275
questions
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2
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359
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Derivative of fractional Laplacian is the fractional Laplacian of the derivative
Is it true that $$\partial_x ((-\Delta)^s u(x)) = ((-\Delta)^s \partial_x
u(x))?$$
user104647
8
votes
1
answer
1k
views
Traces of Sobolev spaces
Is there a simple proof of the following fact?
Theorem. Let $\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain. If $n>2$, then $W^{1,n-1}(\partial\Omega)\subset
W^{1-\frac{1}{n},n}(\...
- 27.3k
1
vote
0
answers
39
views
Production of $H^s$ singularities in the strictly hyperbolic Cauchy problem
This question is a spin-off from Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$ as I am trying to find a solution without ...
- 473
3
votes
1
answer
492
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Hadamard-Rybczynski problem
HR problem deals with a spherical fluid viscous drop falling in a different fluid under influence of gravity. The outer fluid is of uniform speed $U$ in direction of gravity, far away from the drop. ...
- 319
1
vote
1
answer
260
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Recover norm from integral
I am given the following expression where $f \in L^2(\mathbb{R}^2, \mathbb{R}^{2 \times 2})$
$$\int_{\mathbb{R}} \int_{\mathbb{R}} \langle g(x), f(x,y) h(y)\rangle dx dy.$$
The functions $g$ and $h$ ...
user121558
4
votes
0
answers
216
views
Optimal control of SDEs
I've set up a system of stochastic differential equations that I'd like to control. I'm new to optimal control theory and SDEs (and, admittedly, weak on PDEs), so I'm not certain if I've set this up ...
- 41
4
votes
0
answers
607
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Canonical metrics in the context of symplectic geometry?
The investigation of canonical metrics in Kaehler geometry has led to many monumental works in geometric analysis, such as Yau's solution to the Calabi conjecture and more recently, relations between ...
- 975
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votes
0
answers
454
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Measuring the non-commutativity of the codifferential and pullbacks
$\newcommand{\id}{\operatorname{Id}}$
$\newcommand{\TM}{\operatorname{TM}}$
$\newcommand{\Hom}{\operatorname{Hom}}$
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\N}{\...
- 6,621
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0
answers
77
views
Zero energy resonances for scaling critical Schrodinger operators
Given a real valued potential $V\in L^1(\mathbb{R}^3)$, we say that the Schrodinger operator $-\Delta + V$ has a zero-energy resonance if there exists $\psi\in L^2_{loc}(\mathbb{R}^3)\setminus L^2(\...
- 923
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votes
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answers
100
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Regularity of Poisson problem with rough coefficients and mixed boundary conditions
Let $d \in \mathbb N$ and $\Omega \subseteq \mathbb{R}^d$ be open, bounded and connected. Let the boundary $\partial \Omega$ be piecewise Lipschitz and partitioned into a Neumann part $\partial \...
- 171
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vote
0
answers
43
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Hidden regularity for the coupled wave equation with dynamaic boundary condition
We have the equation
\begin{equation}
\left\{
\begin{array}{rrrr}
u_{tt}-\Delta u=0,&\text{in} &
\Omega \times ]0,T[ & \left( 1.1\right) \\
u=0, & \text{on
} & \Gamma _{0}\...
- 545
1
vote
1
answer
119
views
Invariance of the space of harmonic functions under derivation associated to a non-vanishing vector field
Let $X$ be a non-vanishing real analytic vector field on an open manifold $M$. What kind of obstructions would appear when we search for a Riemannian metric on $M$ such that the space of ...
- 312
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votes
0
answers
293
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Spectrum of Laplacian depending on boundary conditions [closed]
Consider a compact domain $\Omega \subset \mathbb{R}^n$ with smooth boundary for simplicity. Consider the Laplacian operator with zero Dirichlet boundary conditions on $\Omega$. It is well-known that $...
- 1,303
11
votes
0
answers
326
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Elliptic regularity of perturbed scalar curvature in Kazdan & Warner
In their paper A Direct Approach to the Determination of Gaussian
and Scalar Curvature Functions, Kazdan and Warner claim something along the lines of: if $g$ is a metric in $W^{2,p}$ ($p>n$) whose ...
- 728
1
vote
1
answer
334
views
Steady Euler flows with compact support?
What is known about (3D) steady incompressible Euler flows with compact support?
(Looking for results in a field you are not familiar with sure is tough.
I had a hope to find clues ...
- 6,861
3
votes
1
answer
727
views
Navier-Stokes equations and machine learning
I am looking for a reference explaining how to solve the Navier-Stokes equations numerically using machine learning algorithms .
Thank you in advance for your help .
- 143
1
vote
1
answer
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Lower bound of the spectrum of a Schrodinger operator on a bounded domain
I am trying to look for references on estimate of the lower bound of the spectrum of a Schrodinger operator $-\Delta + V$ on a bounded domain in three-dimensional space. For simplicity, we can take ...
- 91
2
votes
0
answers
59
views
$L^{1} $ estimate for $2^{nd}$ order elliptic boundary value problem
This is probably a classical question in numerical analysis of PDE (but I don't know the answer).
Suppose you are solving a traditional elliptic problem, for example, $u\in H^1_0(\Omega)$ (in a nice ...
- 2,464
3
votes
1
answer
309
views
Parabolic Regularity with Neumann B.C
Consider the parabolic problem in the cylinder of base $B$, the unit ball,
$$
\partial_t u -\text{div}\left( A(x) D u +F(t,x)\right)=0 \text{ in } (0,T)\times B,
$$
with $(ADu +F)\cdot \nu=0$ on $(0,T)...
- 2,464
7
votes
2
answers
451
views
Do pseudo-differential operators form a sheaf of algebras?
Let $M$ be a smooth manifold.
I have been trying to figure out from the literature I know whether (any flavor of) pseudo-differential operators form a sheaf of algebras (w.r.t. the usual topology on ...
- 7,549
1
vote
0
answers
67
views
Deriving the time evolution of the reflection coefficient for 1d cubic NLS
Update: I have found that the detailed answer to my questions is contained in the book "Solitons: an introduction" by P.G. Drazin and R.S. Johnson. Generally speaking, this seems to be a great book ...
- 163
4
votes
1
answer
1k
views
Existence of solutions to first-order PDE involving convolution
Let $f(x,\alpha)$ be a smooth function of compact support in $x$. Now, let its $\alpha$-dependence be determined by the following first-order equation,
\begin{align}
\frac{\partial}{\partial \alpha} ...
- 193
5
votes
3
answers
1k
views
PDE-oriented textbook on probability and random processes?
I was trained in reaction-diffusion (parabolic/elliptic) PDEs, and my research now focuses on applied optimal tranport. I'd like to learn probability and stochastic processes, mostly their connection ...
- 5,163
3
votes
2
answers
250
views
The gradient $\nabla u$ of $u\in W^{1,p}(M;N)$ is tangent to $N$ almost everywhere
Let $M,N$ be (compact) Riemannian manifolds. $N$ is viewed as an embedded submanifold of $\Bbb R^K$. The Sobolev space $W^{1,p}(M;N)$ is defined as
$$
W^{1,p}(M;N):=\{ u\in W^{1,p}(M;\Bbb R^K)\ |\ u(x)...
- 1,245
3
votes
1
answer
695
views
Looking for access to McKean's original paper?
I'm looking for the PDF version/scan of Henry P McKean Jr.'s paper on propagation of chaos. The reference is as follows -
Propagation of chaos for a class of non-linear parabolic equations., In ...
- 155
1
vote
0
answers
71
views
Von Karman equations system, biharmonic operator,
I am studying the Von Karman equations system (A semi-linear elliptic system of two fourth-order partial differential equations with two independent spatial variables) and I want to solve this ...
3
votes
1
answer
120
views
Meaningful generalization of viscosity solutions to higher order equations
Is there a meaningful generalization of the notion of viscosity solutions to third and fourth order equations?
user75426
2
votes
0
answers
48
views
Proving the existence of solutions of a coupled wave equation with dynamical boundary conditions
I want to prove the existence of the solution of this system by using the Faedo-Galerkin approximation method, I have to choose a basis for working on and I don't know how to do it in this case, I ...
- 545
5
votes
1
answer
441
views
Dependence of the Hölder exponent in De Giorgi-Nash-Moser
I am curious about the Hölder exponent obtained by the De Giorgi-Nash-Moser theory, as a function of the ellipticity.
More precisely: suppose $u$ satisfies weakly
$$
D_i(a^{ij}D_ju)=f
$$
on the $d$-...
- 458
8
votes
0
answers
188
views
Lifting a determinant map
This is a kind of a follow-up to Question on Hessian of a function (probability question). Suppose I give you a continuous function $f:\mathbb{R}^n \to \mathbb{R}.$ Is it true that there exists a ($C^...
- 95.6k
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votes
1
answer
95
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$L^p$ regularity for semidisc
Suppose $B_r\subset \mathbb{R}^2$ is a hemidisc, i.e., $x^2+y^2 \leq r^2, y\geq 0$. Is there a regularity result of the type $\Vert \psi \Vert_{W^{2,p}(B_{1/2})} \leq C (\Vert \psi \Vert_{L^p(B_{1})} +...
- 3,323
2
votes
2
answers
202
views
Iterative method for $p$-Laplacian
Consider the following iterative procedure for solving the $p$-Laplace equation $\nabla \cdot (|\nabla u|^{p-2} \nabla u) = 0$ with fixed Dirichlet boundary data:
$u_0$ is our initial guess, for ...
- 648
3
votes
1
answer
308
views
Typical elements of the space $\mathring {L^k_p}(\Omega)$
In the book Sobolev Spaces with Application of Maz'ya, $\mathring {L^k_p}(\Omega)$ is defined to be the completion of $\mathcal D(\Bbb R^n)$ under the norm $||\nabla^ku||_{L_p(\Omega)}$.
For nice ...
- 1,245
0
votes
1
answer
358
views
Functions satisfying Neumann boundary condition
I have a question about functions satisfying a condition.
Let $D \subset \mathbb{R}^d$ be a Lipschitz domain. That is, for each $x \in \partial D$, there exists an open neighborhood $U$ of $x$ in $\...
- 701
1
vote
0
answers
47
views
Harnack type Estimates for a p-Poisson equation with constant source term
Let $B=B_1(0)\subset \mathbb R^N$ and let $u\geq 0$ solve the PDE
$$
-\Delta_p u=1\,\,\mbox{in $B$}
$$
Let another function $f$ be such that
$$
\begin{cases}
-\Delta_p f =1 \;\;\mbox{in $B$}\\
f=0 \...
- 261
9
votes
1
answer
1k
views
Is there any reason to use paracontrolled calculus over regularity structures?
Paracontrolled calculus was developed by Gubinelli, Imkeller and Perkowski as a way of treating singular stochastic PDEs such as KPZ, $\Phi_3^4$ or PAM, around the same time regularity structures were ...
user69208
7
votes
0
answers
250
views
Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$
I have found the following claim made very clearly at least once in the published literature (see below):
Let $P$ be a linear partial differential operator defined on an open set $\Omega \subset \...
- 473
4
votes
1
answer
244
views
Is this simple-looking weighted poincare-sobolev inequality correct
In Moser's famous paper harnack inequality for parabolic equations, he used the following simple Poincare inequality(Lemma 3)
$\int (f(x)-k)^2 w(x) dx \leq c(w) \int |\nabla f|^2 w(x) dx$
where $k=\...
- 43
1
vote
0
answers
289
views
Harmonic coordinates on asymptotically flat manifold
I am studying the existence of harmonic coordinates at infinity on an asymptotically flat manifold. My Reference papers are, The Mass of Asymptotically Flat Manifold, by Bartnik [B] and The Yamabe ...
- 914
1
vote
0
answers
60
views
Numerical computation of spectrum for operators on real line with "confining potential"
I am looking to understand the conditions under which one can expect "reasonably" accurate solution to leading eigenvalues/eigenvectors of a second order differential operator posed on the real line.
...
- 318
0
votes
1
answer
351
views
Converse of Lax-Milgram theorem [closed]
Suppose that $a(\cdot,\cdot):V \times V \rightarrow \mathbb{R}$ is a symmetric, continuous bilinear form defined on the Hilbert space V.
Assume that, for any continuous linear functional on $l \in V’...
- 23
6
votes
0
answers
265
views
Spectral properties of Non-local Differential operators on real line
I am encountering non-local (and nonlinear) PDEs in my work. To compute stability, I am trying to numerically estimate the spectrum of linearized-but-nonlocal version of the said PDEs.
Definition: A ...
- 318
0
votes
1
answer
102
views
Lipschitz type estimate for the Green function for fractional Laplacian
Let $G(x, y)$ be the Green function of the fractional Laplacian $(-\Delta)^s$ in a bounded interval $I$ of $\mathbb{R}$ with $G(x, y)=0$ on $\mathbb{R}\setminus I$ and $s\in (0, 1).$ Is it possible ...
- 402
3
votes
1
answer
223
views
Particular Elliptic pde of divergence form with indicator boundary data and Feynman-Kac formula
Consider divergence form elliptic pde in smooth boundary domain D
$$ Au:=\sum_{i,j}\partial_{i}(a_{ij}(x)\partial_{i}u(x)), $$
with boundary data $u|_{\partial D}:=1_{A}$ for $A\subset \partial D$. ...
- 4,449
2
votes
0
answers
113
views
Density of $C^0(\Bbb R^{n}\times (0,T))$ and $C^{\infty}_c(\Bbb R^{n}\times (0,T))$ in $L_{p,q}(\Bbb R^n_T)$
Let the space $L_{p,q}(\Bbb R^n_T)$ be defined as the set of all measurable $f:\Bbb R^{n}\times(0,T)\to\Bbb R$ such that $||f||_{p,q}<\infty$, where
$$
||f||_{p,q}:=\left(\int_0^T\left( \int_{\Bbb ...
- 1,245
7
votes
1
answer
452
views
Elliptic operator on compact Hermitian manifold
Let $X^n$ be a compact complex manifold, and $\omega$ be a Hermitian metric on $X$.
Define an operator $P:=i\Lambda_\omega \bar{\partial} \partial$ on the space of the smooth function $C^\infty(X, \...
- 167
4
votes
1
answer
405
views
Question on expansion into Neumann eigenfunctions
Let $\Omega$ be an open bounded domain with a boundary $\partial\Omega$. Consider the following Neumann eigenvalue problem for Laplacian: find $(\phi_n,\lambda_n)\in H^1(\Omega)\times \mathbb{R}$
\...
- 171
-2
votes
2
answers
310
views
$f\in (W^{1,p}(\Omega)\cap C(\Omega) \cap L^{\infty}(\Omega))\setminus C(\bar{\Omega})$, $f=0$ on $\partial \Omega$ imply $f\in W^{1,p}_{0}(\Omega)$?
Q1:
Let $p\geq 1$, and let $f\in W^{1,p}(\Omega)\cap C(\Omega)$. Assume also
$f\in L^{\infty}(\Omega)$ and $f=0$ on $\partial \Omega$. Is it true that
$f\in W^{1,p}_{0}(\Omega)$ even if $f\notin C(\...
- 744
3
votes
1
answer
351
views
Poincare constant on non-convex domain
I'm wondering if there is any results on the optimal constant for the Poincare inequality on a non-convex domain in $\mathbb{R}^3$, since most of the things I found are results on convex ones.
Any ...
- 91
2
votes
1
answer
450
views
Motivation behind the parabolic metric
I've been reading some papers about parabolic evolution problems between manifolds. We want to study the behaviour of maps from the domain $(0,T)\times \Omega$, where $\Omega\subset \Bbb R^m$, to some ...
- 1,245