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.
338
questions
6
votes
2
answers
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On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold
Assume that $M$ is a compact Riemannian manifold whose Laplacian is denoted by $\Delta$. Assume that the Euler characteristic of $M$ is zero. Does $M$ admit a non vanishing vector field ...
5
votes
3
answers
588
views
Higher integrability for Sobolev functions
Let $u \in W^{1,2}(\mathbb{R}^2)$ be a given function satisfying $$\frac{1}{|B_r|}\int_{B_r(x)} |\nabla u|^2 dy \leq \frac{1}{r^{\delta}}$$ for all $r \leq 1$, $x \in \mathbb{R}^2$ and some fixed $\...
5
votes
2
answers
1k
views
Analytic solution of a system of linear, hyperbolic, first order, partial differential equations
In a try to solve a physical problem, I've faced a system of first-order partial differential equations of the form
$$\cos\left(t\right)\partial_{x}\mathbf{u}+\sin\left(t\right)\partial_{y}\mathbf{u}+...
5
votes
2
answers
397
views
Backward heat equation and forward perturbed heat equation well posed?
I consider the following scenario. Let $I$ be a compact interval in space and $f$ a nice function in the space $C^{\infty}(I)$. In the following we consider a self-adjoint realization of our operators ...
4
votes
1
answer
294
views
The behavior of $ \nabla u $ on the boundary for Poisson equations
Let $ \Omega $ be a bounded domain with smooth boundary. Consider the Poisson equation
\begin{eqnarray}
-\Delta u&=&f\text{ in }\Omega\\
u&=&0\text{ on }\partial\Omega
\end{eqnarray}
...
4
votes
0
answers
183
views
improved regularization for $\lambda$-convex gradient flows
It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...
4
votes
2
answers
2k
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System of linear first order PDE with constant coefficients
recently in my researches I've come across the following operator
$$L\left(\begin{array}{c}
a_1\\
\vdots\\
a_n
\end{array}\right)=M_1\left(\begin{array}{c}
...
4
votes
1
answer
591
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
$$\...
4
votes
1
answer
200
views
Mapping properties of backward and forward heat equation
In a previous question on mathoverflow, I asked about the following:
Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions.
The ...
3
votes
2
answers
626
views
A function in $W^{1,p}(\Omega)$ for $1 < p < n$ which is not differentiable a.e
I'm seeking a function which belongs to $W^{1,p}(\Omega)$ for $p < n$ which is not differentiable a.e. There is a standard theorem which shows that if $p > n$ then in fact any function in $W^{1,...
3
votes
3
answers
1k
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Uniqueness of solution of the wave equation
Consider the wave equation
$$\frac{\partial^2 u}{\partial t^2}-\sum_{i=1}^n\frac{\partial^2 u}{\partial x_i^2}=0$$
with initial conditions
$$u|_{t=0}=\frac{\partial u}{\partial t}|_{t=0}=0$$
Does ...
3
votes
1
answer
5k
views
About eigen-functions of the Gaussian kernel
If I look at the Guassian kernel function $e^{- \frac {\vert x - y\vert_2^2 }{2 w^2 } }$ for $x, y \in \mathbb{R}$. Then w.r.t the Gaussian measure $N(\mu,\sigma)$ I believe it is true that this has a ...
3
votes
2
answers
182
views
Bounding integral expression with total variation of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{(g(x)-g(y))(x-y)}{|x-y|^{3}} d x d y $$
for $\epsilon>0$, $f \in L^\infty(\mathbb R)$,...
3
votes
1
answer
307
views
Typical elements of the space $\mathring {L^k_p}(\Omega)$
In the book Sobolev Spaces with Application of Maz'ya, $\mathring {L^k_p}(\Omega)$ is defined to be the completion of $\mathcal D(\Bbb R^n)$ under the norm $||\nabla^ku||_{L_p(\Omega)}$.
For nice ...
3
votes
1
answer
988
views
Simple existence and uniqueness for second order and linear elliptic PDE
Consider a closed Riemannian manifold $(M,g)$ and let $u \in C^{2,\alpha}(M)$ be a positive function on $M$.
I am interested on the existence of solution for the following problem: given a continuous ...
3
votes
1
answer
2k
views
Method of characteristics of a system of first order pdes
I asked the question on math.stackexchange.com, but didn't get any reply. So, I asked it again here. Any suggestion or hint is welcome, and thank you for your attention.
Consider the system of first ...
3
votes
2
answers
868
views
Representing a nonlinear elliptic PDE as an energy minimization problem
I need to solve a PDE in 2D representing a (time-independent) nonlinear diffusion process. The unknown function is $\phi(x,y)$ and its gradients create fluxes $\vec J$ through a nonlinear relation:
$$\...
2
votes
0
answers
225
views
Dense property of intersection of Sobolev space
I'm using Muscalu and Schlag's textbook (online notes) to study Littlewood-Paley theory in harmonic analysis, where I encounter the following claim:
Pick an arbitrary real number $s$, we have that the ...
2
votes
1
answer
315
views
The study of dynamics of a polynomial vector field via Green's function methods
In the litterature, in particular in the papers on dynamical investigation of polynomial vector fields on the plane, are there some research devoting to study the Green's function for the PDE which is ...
2
votes
1
answer
309
views
Gevrey estimate of derivatives
Let $\phi$ be the $C^\infty$ function defined on $\mathbb R_+^*$ by $e^{-t^{-2}}$ and by $0$ on $\mathbb R_-$.
Question: I think that there exists $\rho>0$ such that
$$
\forall t\in \mathbb R,\...
2
votes
0
answers
206
views
Variational formulation for elliptic interface problem
Where can I find a paper that deals with the following interface problem with variational methods? In particular, what is the correct variational formulation of the problem (that is, a functional ...
2
votes
1
answer
723
views
Existence of a solution to an infinite dimensional Stratonovich SDE
Let
$U,H$ be separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with finite trace
$U_0:=Q^{1/2}U$
$(\Omega,\mathcal A,(\mathcal F_t)_{t\ge 0},\operatorname P)$ ...
2
votes
1
answer
923
views
Derivative and Jacobian determinant of solution of ODE [closed]
Let $\Phi$ be the unique solution of
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{cases}$$
where we have assumed $f$ smooth.
...
2
votes
0
answers
111
views
Bounding integral expression with BV norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
1
vote
2
answers
465
views
Non-closed range space of Laplace operators?
Set $ -\Delta: H^2(\mathbb{R}^3) \subseteq L^2(\mathbb{R}^3) \to L^2(\mathbb{R}^3) $. Then $ \mathcal{R}(-\Delta) $ is non-closed?
Sorry if this question is trivial. I am not familiar with theory of ...
1
vote
2
answers
549
views
Prove Liouville theorem without using mean value property
How can I prove the following Liouville theorem without using the mean value property?
If $u$ is harmonic on $\mathbb{R}^n$ and $\int_{\mathbb{R}^n}|\nabla u|^2 dx \leq C$ for some $C > 0$, then $...
1
vote
0
answers
70
views
Derivation of the vortex filament equation from Euler equation
How can the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$,
be derived from the Euler equation
$$\partial_t \...
1
vote
1
answer
231
views
Elliptic interface problem without conditions on the interface
Consider an open domain $U$ split in two non-overlapping subdomains: $U = U_1 \cup U_2$.
For a model case, consider a ball split in a smaller ball and an anulus.
Consider the following elliptic ...
1
vote
1
answer
636
views
Properties of the trace term in the Itō formula
Let's consider the SDE $${\rm d}X_t=u_t(X_t){\rm d}t+\xi_t(X_t){\rm d}W_t\;\;\;\text{for all }t\ge 0\tag 1$$ where
$U,H$ are separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ is nonnegative ...
107
votes
8
answers
15k
views
What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?
I know the following facts. (Don't assume I know much more than the following facts.)
The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem.
The ...
75
votes
13
answers
8k
views
Counterexamples in PDE
Let us compile a list of counterexamples in PDE, similar in spirit to the books Counterexamples in topology and Counterexamples in analysis. Eventually I plan to type up the examples with their ...
58
votes
9
answers
9k
views
Motivation for and history of pseudo-differential operators
Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds). Which prominent example problems lead you to work with pseudo-differential ...
32
votes
1
answer
5k
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Jet bundles and partial differential operators
A geometric way of looking at differential equations
In the literature for the h-principle (for example Gromov's Partial differential relations or Eliashberg and Mishachev's Introduction to the h-...
31
votes
5
answers
4k
views
Is there a mathematically precise definition of turbulence for solutions of Navier-Stokes?
Given a solution $S$ of the Navier-Stokes equations, is there a way to make mathematically precise a statement like: "$S$ is turbulent in the spacetime region $U$"?
And if such a definition ...
29
votes
1
answer
4k
views
What goes wrong for the Sobolev embeddings at $k=n/p$?
For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the celebrated Sobolev inequalities:
If $k < n/p$ then $u\in L^q(U)$ for $q$ satisfying $\frac{...
29
votes
18
answers
3k
views
PDEs as a tool in other domains in mathematics
According to the large number of paper cited in MathSciNet database, Partial Differential Equations (PDEs) is an important topic of its own. Needless to say, it is an extremely useful tool for natural ...
28
votes
4
answers
6k
views
Is there any way to rewrite a partial differential equation using language of differential forms, tensors, etc?
My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there any way (or any algorithm) ...
25
votes
5
answers
5k
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Book Recommendation - PDE's for geometricians / topologists
I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as ...
25
votes
1
answer
2k
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The origin of Discrete `Liouville's theorem'
It is known that discrete Liouville's theorem for harmonic functions on $\mathbb{Z}^2$ was proved by Heilbronn (On discrete harmonic functions. - Proc. Camb. Philos. Soc. , 1949, 45, 194-206).
If ...
25
votes
8
answers
9k
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Applications of PDE in mathematical subjects other than geometry & topology
Partial differential equations have been used to establish fundamental results in mathematics such as the uniformization theorem, Hodge-deRham theory, the Nash embedding theorem, the Calabi-Yau ...
24
votes
3
answers
3k
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Does elliptic regularity guarantee analytic solutions?
Let $D$ be an elliptic operator on $\mathbb{R}^n$ with real analytic coefficients. Must its solutions also be real analytic? If not, are there any helpful supplementary assumptions? Standard ...
22
votes
2
answers
7k
views
What's the idea behind Carleman estimates?
A standard Carleman-type estimate is of the form
$$
\sum_{|\alpha|<m}{\tau^{2(m-|\alpha|-1)}\int{|D^{\alpha}u|^{2}e^{2\tau\phi}}dx}\leq K\int{|Pu|^{2}e^{2\tau\phi}dx},\quad u\in C_{0}^{\infty},
$$
...
21
votes
0
answers
2k
views
Characterising critical points of $E(f)=\int_{M}| \bigwedge^2 df|^2 \text{Vol}_{M}$
$\newcommand{\id}{\operatorname{Id}}$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\TM}{\operatorname{TM}}$
$\newcommand{\Hom}{\operatorname{Hom}}$
$\newcommand{\Cof}{\operatorname{Cof}}$
$\newcommand{\...
21
votes
1
answer
4k
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Image of the trace operator
It is well-known that we have the trace theorem for Sobolev spaces. Let $\Omega$ be an open domain with smooth boundary, we know that the map
$$ T: C^1(\bar\Omega) \to C^1(\partial\Omega) \subset L^...
21
votes
2
answers
5k
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Elliptic regularity for the Neumann problem
I'm trying to understand how to establish regularity for elliptic equations on bounded domains with Neumann data.
For simplicity, let's presume we are focusing on $-\Delta u = f$ in $\Omega$ and $\...
18
votes
3
answers
2k
views
Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order derivational operators
EDIT: According to some comments on this post I revise the title to remove the misunderestanding.
Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated ...
18
votes
4
answers
6k
views
unique continuation principle
I recently encountered a paper by Protter ("Unique continuation of elliptic equations") that starts out by saying "any solution of an elliptic equation that is defined on a domain $D$ must vanish on ...
17
votes
6
answers
2k
views
Can the "physical argument" for the existence of a solution to Dirichlet's problem be made into an actual proof?
Caveat: I don't really know anything about PDEs, so this question might not make sense.
In complex analysis class we've been learning about the solution to Dirichlet's problem for the Laplace ...
16
votes
1
answer
930
views
Are isospectral manifolds necessarily homeomorphic?
It's known that there are pairs of closed Riemannian manifolds which are isospectral but not isometric.
Is it known if there are closed Riemannian manifolds which are isospectral but not homeomorphic?...
15
votes
1
answer
5k
views
Basis for the space of Harmonic homogeneous polynomial in N variables.
Hello,
Does someone know an explicit basis of the space of harmonic homogeneous polynomial in N variables.
When $N=3$, if I'm not mistaking Legendre polynomial allow to write an explicit basis.
Is ...