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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optimal regularity for Laplace equation with inhomogeneous L^p Robin boundary condition
Consider the problem
$$-\Delta u = 0 \mbox{ in }\Omega,\qquad \partial_\nu u+\tau u=g\mbox{ on }\partial\Omega,$$
where $\Omega\subset R^n$ is a bounded $C^2$-domain, $\tau>0$ is a constant, and $...
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nodal lines in the dirichlet problem
In the Dirichlet problem if nodal lines do not touch $\partial\Omega$ (unit disk), what happens to the eigenvalues?
Thanks for help.
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null controllability of linear wave equation
Consider the linear wave equation :
$$z_{tt}=\Delta z + k(x) z + h(t) , \; in \; \Omega\times (0,T)$$
Are there sufficient conditions on the functions $k(x)$ and $h(t)$ for which $(z,z_t)$ vanish ...
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Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism
The variational bicomplex seams to provide a modern formulation of the variational problem in terms of modern differential geometry. In particular the bigraded complex of differential forms $\Omega^{p,...
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Existence of a function
[also asked here http://math.stackexchange.com/questions/307197]
All arguments are in $\mathbb{R}^3$.
Suppose $n(x)$ is a smooth function where $\mathbf{supp}(n(x)-1)$ is a compact set $\Omega$. i....
CommunityBot
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1
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What is visualization of gradient flow of a functional?
I don't work on functional analysis but during my study, I faced gradient of a functional. I read its definition, but I can not understand why it is a useful tool? Why if a flow can be written as a ...
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About definition of weak derivative in abstract PDE problems
I'm confused about weak derivative definition.
$u \in L^2(0,T;V)$ has weak derivative $u'\in L^2(0,T;V')$ iff
$$\int_0^T u(t)\varphi'(t) = -\int_0^T u'(t)\varphi(t)$$
holds for all $\varphi \in C_0^...
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strong stability for the wave equation
Consider the $n-$dimensional wave equation
$$z_{tt}=\Delta z + k(x) z - \epsilon {1}_\omega z_t, \; in \; \Omega\times (0,T)$$
where $\omega\subset \Omega.$ Can I have $z(t) \to 0,$ as $t\to+\infty$ ...
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Representing immersions from a surface into 3-space
Let $\mathbb T^2=(S^1)^2$ be the 2-torus, for convenience. $\def\Imm{\operatorname{Imm}}$
Consider the Frechet manifold of immersion $\Imm(\mathbb T^2, \mathbb R^3)$ and the smooth mapping
$R:\Imm(\...
- 108k
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Asymptotic decay for the wave equation
Consider the wave equation
$$
y_{tt} = \Delta y - \epsilon y_t
$$
on $\Omega\subset R^n$, with Dirichlet boundary conditions.
Where $\epsilon >0$.
Is it possible to find an explicit value $\...
- 39k
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Analyticity of the solutions of PDE
Let's consider a (partial) differential equation with analytic coefficients. The initial conditions may be non-analytic*.
Is it possible a solution $f$, which is non-analytic at any point of the ...
CommunityBot
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Density of smooth functions in Sobolev spaces on manifolds
Hebbey defines the Sobolev space of functions on a Riemannian manifold (M,g) as the completion of smooth functions under the Sobolev norm. However, I have seen (elsewhere) that Sobolev spaces have ...
- 25.3k
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Analytical solution to a Linear advection-reaction PDE
I am looking for an analytical solution for the linear PDE
$(1)\qquad\qquad \qquad f_t+ A f_x + B f = 0, $
Where $A$ and $B$ are constant matrices and $f=f(x,t)$ is a vector.
Clearly each one of $...
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1
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Is there a lower bound for variance in terms of curvature?
If the Gaussian curvature of the metric $g= f^2(x,y)(dx^2+dy^2)$ is nonzero then $f$ cannot be constant. This can be expressed by stating that the (probabilistic) variance $Var(f)$ of $f$ is nonzero (...
- 16.6k
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Nonharmonic solutions of Laplace's equation
Let $f \colon U \to \mathbb{R}$ be a twice differentiable function, where $U$ is an open subset of $\mathbb{R}^n$. Here twice differentiable means that all the second partial derivatives $\frac{\...
- 17.1k
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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 p\...
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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.
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Harnack's Inequality and (hypo)elliptic PDE
Background: I am aware of the Harnack's Inequality for linear elliptic equations.
My questions are:
(a) Is there a version of Harnack's Inequality for nonlinear elliptic equations, say, of the form ...
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Sharpness of the Sobolev embedding theorem
We know that $W^{k,p}\hookrightarrow C^{k-\lfloor\frac{n}{p}\rfloor-1,\gamma}(\bar{\Omega})$ with $kp>n,\gamma=\lfloor\frac{n}{p}\rfloor+1-\frac{n}{p}$, where $n$ is the dimension of $\Omega$, $\...
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1
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What we mean by positive solution and radial solution of any partial differential equation
I am not getting the most of the article concern with the existence and uniqueness of positive solution and radial solution.
I just want to know how a positive solution and radial solution of ...
- 648
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Manifold structure for the set of solutions to a first order elliptic system?
Consider a bounded domain $S\subset R^2$ and an elliptic system of two first order PDEs, namely a generalization of the Cauchy-Riemann system allowing nonconstant coefficients and lower order terms. ...
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Trace Inequality question
There is a result in a paper I am reading :
Let $\Omega$ be a bounded domain. For any $\epsilon > 0$, there is a constant $C(\epsilon)$ such that
$$\lVert n \times u\rVert_{H^{-1/2}(\partial \...
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1
answer
275
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On the fractional Schrödinger equation
I wonder if there is any theory about what we can call the fractional Schrödinger equation:
$$
\mathrm{i}\frac{\partial \psi}{\partial t} = (-\Delta)^s \psi + g(|\psi|^2)\psi \quad\hbox{in $\mathbb{R}^...
- 188k
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1
answer
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Does $\partial\overline{\partial}f=0$ imply $f\equiv c$ for particular kind of $f$?
Hi!
Let $f\in C^{2,\alpha}\left( \mathbb{C}^{m}\setminus \overline{B_{R}},\mathbb{R} \right)$ with $m\geq 2$, $R>0$ and s.t. $f$ has an expansion of type
$$f=1+\mathcal{O}\left( \frac{1}{|z|} \...
- 108k
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Galerkin method for existence for PDE with nonsymmetric bilinear form
Suppose we have a PDE
$$\langle u', v \rangle + a(u,v) = 0$$
where $a:V\times V \to \mathbb{R}$ is a bounded symmetric bilinear form, then if $u_0 \in V$ then $u \in L^2(0,T; V)$ with $u' \in L^2(0,T;...
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Integral of a harmonic function on a manifold with two non-parabolic ends
Let M be a complete Riemannian manifold.Suppose there are two non-parabolic ends on M with respect to $M\backslash {B_p}\left( {{R_0}} \right)$Then there is a harmonic function f on M.Is it right that ...
- 17k
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Is there a good way to estimate the Fourier transform of $\frac{1}{\lambda-iP(\xi)}$
Assume that P is a real valued strong elliptic polynomial, then what do we know about the following
$$
K(\lambda,x)=\int{\frac{e^{ix\xi}}{\lambda-iP(\xi)}}d\xi,\quad \lambda\in \mathbb{R}\0
$$
The ...
CommunityBot
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What goes wrong for the Sobolev embeddings at $k=n/p$?
For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the celebrated Sobolev inequalities:
If $k < n/p$ then $u\in L^q(U)$ for $q$ satisfying $\frac{...
- 4,899
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answers
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Extension divergence-free, curl-converging vector field
Hi.
Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole ...
- 3,425
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Orthogonal projection of discontinuous piecewise polynomial space in energy scalar product
Let $I = [0,1]$ be the unit interval Let $I$ be partioned into $n$ closed subintervals $(I_j)_J$, each of length $1/n$.
Let $X_{DC} = \{ v \in L^2[0,1] | 1 \leq j \leq n : v_{|I_j} \in \mathcal P_1( ...
- 5,327
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what's the motivation of Weyl calculus ?
In the pseudo-differential operator theory, we can define a pseudo-differential operator by $$a(x,D)u=(2\pi)^{-n}\int{a(x,\xi)e^{i\langle x-y,\xi \rangle}u(y)dyd\xi}$$ with $a(x,\xi)$ belong to some ...
CommunityBot
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Splitting the action of functionals in duals of Sobolev spaces
Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
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How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$?
How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$?
- 11.2k
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well-posedness of the transport equation
I asked this question before on math exchange but did not have any luck with an answer. I would like to consider a simple example but get a thorough understanding of the theory behind it. I am ...
CommunityBot
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Boundedness of a given boundary value problem.
I've been given the following BVP:
\begin{align*}
-\Delta u = u- u^3,\: x\in \Omega
\end{align*}\begin{align}
u = 0,\: x\in \partial \Omega
\end{align}
where $\Omega\subset \mathbb{R}^N$ is bounded.
...
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Eigenfunctions of the laplace on the 2-sphere with conformal metric induced by schwarzschild
Hi,
it's well known that the coordinates $x_1$, $x_2$, $x_3$ are the first three eigenfunctions with positiv eigenvalues ($=2/r^2$) of the negative laplace-beltrami operator ${-}\triangle$ on the 2-...
CommunityBot
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A 'conjecture' on critical elliptic pde
I conjecture the following.
Let $\Omega=\mathbb{R}^3-\overline{B_1(0)}$. Define
$$E_{\Omega}(u)=\frac{1}{2}\int_{\Omega}|\nabla u|^2dx-\frac{1}{6}\int_{\Omega}|u|^6dx.$$
$E_{\mathbb{R}^3}$ is defined ...
- 539
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"Integration by parts" formula for functionals
We know that for a Hilbert triple $V \subset H \subset V^{'}$, if we have $u, v \in L^2(0,T;V)$ with $u',v' \in L^2(0,T;V')$
then
$$\frac{d}{dt}(u(t), v(t))_H = u'(t)(v(t)) + v'(t)(u(t))$$
where the $...
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1
answer
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Norm of differential operator between Sobolev spaces
It is easy to check that the differential operator $\partial^a$ (where $\alpha\in \mathbb{N}_0^n$) is continuous between the Sobolev spaces (with usual norms)
$W^{m,p}(U)\to W^{m-|\alpha|,p}(U)$, ...
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Spectrum of Combinatorial Laplacian
The spectrum of the combinatorial laplacian is well understood for a square lattice. What about for other lattices?
In particular:
Let $ f: \mathbb{Z}^2 \rightarrow \mathbb{R} $. The usual ...
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Viscosity solution of the PDE
Let $\Omega$ be bounded domain, $u=0$ on $\delta\Omega$ and
$$|Du|-f(x,u)=0$$
where $f\ge 0$ and $f$ is strictly monotone for fixed $x.$ I am looking for the reference to show that it has unique ...
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3
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1
answer
354
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Solvability for constant-coefficient partial differential operators
Let $\mathcal{S}$ denote the space of Schwartz functions on $\mathbb{R}^n$, and $\mathcal{S}'$ the space of tempered distributions. Let $L$ denote a linear, constant-coefficient, partial differential ...
- 16.2k
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Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform II
This is a modification of a previous question.
Question: Suppose $a(s)\in C^\infty([0,1])$ and $H(s,x)\in C^\infty([0,1]\times [0,1])$ with $H(s,x)>0$, $\forall s,x\in [0,1]$. Suppose,
$$\sup_{\...
CommunityBot
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0
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Looking for higher order Sobolev inequality
Hello,
On a compact (without boundary) Riemannian manifold (eg. some surface in $\mathbb{R}^n$), I'm looking for a result like
$$\lVert \nabla u\rVert_{L^2}^2 \leq \epsilon\lVert u\rVert_{H^1}^2+\...
- 29
15
votes
1
answer
858
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Symbols of elliptic operators
First let me state the problem, then I'll explain its origin and finally, I'll ask the main question..
Problem S. Fix a positive integer $n$. Find all the pairs $(V, S)$, whith the following ...
- 34.7k
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0
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A critical elliptic PDE
I am considering the problem $-\Delta u=|u|^4u$, $x\in \Omega\subset \mathbb{R}^3$, $u|_{\partial \Omega}=0$. Where $\Omega$ is a unbounded domain. Some special case like $\Omega=\mathbb{R}^3-B_1(0)$, ...
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1
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479
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Existence of solution for this parabolic PDE
The parabolic PDE
$$\langle u', v \rangle + a(u,v) = \langle f, v \rangle$$
has a unique solution $u \in L^2(0,T; H^1)$ with $u' \in L^2(0,T;H^{-1})$ if $a$ is a bounded and coercive bilinear form (...
CommunityBot
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2
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0
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Best constant of Gagliardo-Nirenberg inequality in exterier domain
In $\mathbb{R}^N$, we know that the best constant of Gagliardo-Nirenberg is characterized by the solution $Q$ of $-\Delta u+u=|u|^2u$ with minimal mass. One have
$||u||_4^4\leq C||u||_2||\nabla u||_2^...
- 52.3k
2
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2
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397
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how many nonparabolic ends guarantee a nonconstant harmonic function on Riemannian manifold?
M is a Riemannian manifold.An end E is said to be a non-parabolic end if it admits a positive Green's function with Neumann boundary conditon on E.Otherwise,it is a parabolic end.If M has more than ...
- 4,131
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2
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326
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Are $\lVert \Delta u \rVert_{L^2(S)}$ and $\lVert u \rVert_{H^2(S)}$ equivalent norms on a compact manifold?
Hi,
I am looking for the result:
$$\text{The norm} \quad \lVert \Delta u \rVert_{L^2(S)} \quad \text{is equivalent to} \quad \lVert u \rVert_{H^2(S)}$$
for scalar functions $u \in H^2(S)$, where $S$ ...
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