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.
4,466 questions
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Examples of functions in $W^{k,p}(\Omega)$ with exact smoothness
Please give, explicitly, a function $f:\Omega\mapsto\mathbb{R}$ such that $f\in W^{k,p}(\Omega)$ but $f\notin W^{s,p}(\Omega)$ for $s>k$.
Here $\Omega$ can be a subset of $\mathbb{R}^n$ with ...
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One parameter family differentiable dependence for linear parabolic pde's
Consider for example, the Black Schole's equation
$$
\partial_tu+0.5\sigma^2s^2\partial_{ss}u+rs\partial_su-ru=0
$$
on $[0,T]\times[0,\infty)$ subject to boundary conditions $u(s,T)=f(s)$.
The ...
- 383
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K-Theory of Algebra of Zeroth Order Pseudo differential operators
Any one knows a reference for computing K_0 of Algebra of zeroth order Pseudo's on a closed manifold in terms of explicit generators?
Thanx!
- 309
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How fast does the Heat equation with boundary condition $\frac{\partial u}{\partial \vec{n}}=u^2$ decay?
Consider the heat equation $\frac{\partial u}{\partial t}=\frac{1}{2}\Delta u$ in a bounded domain (say the interval [0,$\pi$]) with boundary condition $$\frac{\partial u}{\partial \vec{n}}=u^2$$
with ...
- 165
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One-parameter group of unitary operators and Core
Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...
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Reference to the Existence and Uniqueness of the PDE system
I've the following Problem on systems of Partial Differential Equations. I have "$ N $" Physical variables. and Finally I form the equation on a bounded domain having regular boundary in $R^d$ ($d=2$...
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90
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Monotone operator subgradient
I am trying to solve a PDE of the form $\mathcal{A}u'(t) + \partial\Psi[u(t)] \ni 0$ where $\mathcal{A}$ is a skew-symmetric, linear, monotone operator, $\Psi$ is convex, and $\partial \Psi$ is the ...
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A question about PDE argument involving monotone convergence theorem and Sobolev space
I'm reading this paper. In it there is the following argument (see page 240).
Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. $b(\...
- 266
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Is $\Delta u+f\in (H^1(\Omega))^*$ with $u\in H^1_0(\Omega)$ and $f\in L^2(\Omega)$?
Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $\partial \Omega$ being $C^2$. Suppose $u\in H^1_0(\Omega)$, $f\in L^2(\Omega)$ and $\nu>0$. It is said in the Navier-Stokes Equations ...
user14319
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Finite propagation speed of second order operators
I was reading about the finite propagation speed of the wave equation on a Riemannian manifold. I was wondering if instead of the Laplace-Beltrami operator $\Delta$ we consider the equation $$\...
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$f,g , |f|f, |g|g \in A(\mathbb R) \ \text{(Banach algebra)} \implies \left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$?
Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra
$$A(\mathbb R):= \{f\in L^{1}(...
- 1,051
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Geometric Mean Value Property
Does anyone know where I could find a proof of a variant of a version of the mean-value property for harmonic functions in Riemannian manifolds? I'm actually more interested in using an elliptic ...
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The continuity of $L^2$ gradient on moving domain
I post this on MSE too. Sorry for this short cross posting. But I stuck in this problem for a while... I don't know what the right tool should be used to handle the moving domain problem...
Let $I:=(...
- 679
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Is Poisson's kernel integrable?
Let $E$ be a smooth domain. Green's function is defined as $G(x,y)=F(x,y)-\Phi(x,y)$ where $F$ is the fundamental solution to the Laplace equation. For a fixed $x\in E$, $\Phi(x,\cdot)$ is a harmonic ...
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Solutions to the wave equation on non orientable surfaces like a mobius strip
Given a mobius strip, what do the solutions of the wave equation look like qualitatively? How do they differ from solutions on the equivalent strip glued together as a cylinder? Any refs, ...
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Analysis of Sobolev spaces [closed]
I just wanted to know wthether the following is OK or not.
Let $X$ be $H_0^1(\Omega)\bigcap L^{\infty}(\Omega)$, thought of as a subspace of $H^1_0(\Omega)$ and endowed solely with the usual $H^1$ ...
- 157
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existence of an initial-boundary value problem with nonhomogeneous boundary conditions
Let $n\geq 2$ be an integer and $\Omega\in R^n$ be a bounded domain with boundary $\partial\Omega$ . Consider the following IBVP:
$u_t=\Delta u$, for $x\in \Omega$, $t>0$;
$u(x, 0)=f(x), x\in\...
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Dichotomy for global existence or blow up for solutions of evolution problems
Consider the problem (Nonlinear Schrödinger equation)
\begin{equation}
\left\{
\begin{array}{rl}
iu_t + \Delta u\mp u|u|^{\alpha}=0\\
u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\
\end{array}\...
- 403
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I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection
I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
user74319
3
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Equivalence of distributional and viscosity solution in parabolic case?
In the answer to https://math.stackexchange.com/questions/166286/viscosity-solution-vs-weak-solution
H. Ishii, "On the equivalence of two notions of weak solutions, viscosity solutions and ...
- 383
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Can we expect $\left\||f|^{2}f-|g|^{2}g\right\|\leq C ||f-g||$ in the Banach algebra $A(\mathbb T)$ ?
Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space,
$$A(\mathbb T):= \{f\in L^...
- 1,051
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Comparison principle for partial differential equation with singular coefficients
How (or if) a comparison principle works in the case of equations
singular at some point? For example, I am analyzing a partial
differential equation
$$
\partial_{t}u=\partial_{rr}u+\frac{2}{r}\...
- 263
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3
answers
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Laplace equation with mixed boundary conditions
Does the Laplace equation on a rectangle with Dirichlet boundary conditions at two opposing sides and Neumann boundary conditions at the other two, always have a solution? If it does, is it unique? Is ...
- 221
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Prescribed curvature problem of a connection beyond the real analytic category for $SL(3,R)$ bundles?
With reference to the questions Does the Riemann-Christoffel curvature determine the connection? and When is a given matrix of two forms a curvature form?, and recalling the following important result ...
- 573
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Sobolev spaces defined on non-compact Lie groups
In this post, a question was raised to discuss the generalization of Sobolev spaces on locally compact Lie groups. Now my question is whether there exists a generalization of Sobolev spaces and ...
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A closed extension of the Laplace operator with respect to the supremum norm
Let $X$ be a bounded connected open subset of the $n$-dimensional real euclidean space. Consider the Laplace operator defined on the space of infinitely differentiable functions with compact support ...
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Boundary behaviour of a second order pde with characteristics
Good morning everybody. My question is inspired from the following fact:
Consider $\mathbb R^3$ endowed with coordinates $(x,y,z)$. Of course if we were to solve the second order pde $\partial_x^2 g(...
- 277
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Rellich-Necas identity
I am looking for a book/paper which has the proof of the Rellich-Nicas identity.
[EDIT by Yemon Choi] It seems that what was meant is "the Rellich-Necas identity", although the original poster hasn't ...
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Motivation for some operators in the dyadic model of Navier Stokes equation
What's the motivation for cascade operator $C_{u}, C_{d}$ (and then the dyadic version of operator $B=(u\cdot \nabla)u$), which is
$(C_{u}(u,v))_{Q} = 2^{\frac{5}{2}j}u_{\tilde{Q}}v_{\tilde{Q}}$,
$(...
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System of first order PDE
I have a system of first-order nonlinear partial differential equations.
$$A(x,t)\frac{\partial u}{\partial t}(x,t) + B(x,t)\frac{\partial u}{\partial x}(x,t) + c(x,t) = 0$$
$$x \in \mathbb{R}, \quad ...
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$\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$?
Let M be a complete Riemannian manifold.If there exists a positive function defined on M satisfying$\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$?
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Equations for Integrable Systems
So, let's say we have a symplectic variety over $\mathbb{C}$, $M$, of dimension $2n$, and $f_1,\ldots,f_n$ Poisson commuting functions with $df_1\wedge\ldots\wedge df_n$ generically nonzero. Further ...
- 16k
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Elliptic Harnack inequality for 1D Schrodinger operator?
For a nonnegative polynomial $V: \mathbb{R} \to \mathbb{R}$, write $H = -\Delta + V$. I am wondering if there is an elliptic Harnack inequality for H. That is:
There exist $C_{H} > 0$ and $\delta \...
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Theorem with an example [closed]
i have this theorem
in the paper they gives an example:
but here $H_1$ is not satisfied !
How to correct it please?
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Please recommend some literature on the systematical theory of the elliptic systems!
Now I'm interested in the theory of elliptic systems, for example, both the linear and nonlinear case, the exsitence and regularity results, and is there a Fredholm alternative result for the linear ...
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Failure of Fredholm property of elliptic PDE systems
Roughly speaking, a PDE operator satisfies the Fredholm property if its principal symbol is elliptic and the information provided on the boundary satisfies the Shapiro-Lopatinskii condition.
What can ...
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Well-posedness of heat equation with distributional right hand side
The question is about well-posedness of heat equation
$$
\frac{\partial\Theta}{\partial t}=\alpha^2\Delta\Theta+p(t)\delta(x-u(t))\delta(y-v(t)),~~ (x,y,t)\in\Omega\times[0,T],
$$
subjected to ...
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1
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Elliptic Differential Equations with rough boundary data
Question stated roughly:
Consider the Dirichlet problem for an elliptic equation on a ball. How much can we say about regularity at the boundary of non-linear elliptic equations? Further, how can one ...
- 469
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Constant in a trace Sobolev theorem for concave domains
I wonder is the following inequality is true/known:
Let $\Omega\subset \mathbb{R}^n$ be a (locally) Lipschitz domain which is the complement of a convex set, then
$$
\int_{\partial\Omega} |u|^2 ds
\...
- 175
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238
views
Schauder estimate on a bounded domain
We know if $u:\mathbb{R}^{n}\to \mathbb{R}$ is a smooth function with compact support, then we can show, via integration by part, that
$$
||\Delta u||=||{\nabla}^2u||
$$
where $||\cdot||$ is the $L^2(\...
- 81
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3
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What are the basis functions for a product space?
Let $X=L^1\left([0,1]^3\right)$,
for numerical purpose, what are the possible basis function for $X$?
In finite element method, the basis functions are tooth functions, or polynomial functions.
Is ...
user24451
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2
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635
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vector valued pde's good reference
I recently came across a Dirichlet problem for a vector valued functions. In broad terms the problem is as follows.
Suppose $\Omega \subset \Bbb R^n$ is a smooth bounded domain, $P:C^\infty(X)^n \...
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Nonlinear Schrödinger blow-up for non radial solutions
I am studying a paper of Frank Merle and Pierre Raphaël,
http://math.unice.fr/~praphael/Publications/blow-up-norme-critique.pdf.
The equations are
$$
i\partial_tu+\Delta u=-|u|^{p-1}u
$$
on $\mathbb{R}...
- 1,616
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$W^{2,p}$ or $W^{1,q}$ regularity for the laplace on a euclidean sphere
Hi,
it is easy to prove the $W^{2,2}(\mathcal S^2)$ regularity for the laplace on the (2 dimensional-) standard sphere $\mathcal S^2:=\lbrace x \in\mathbb R^3: \vert x\vert=1 \rbrace\hookrightarrow\...
3
votes
1
answer
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views
Trace theorem for manifolds with boundary
Can I get a reference to a trace theorem for a manifold $M$ with boundary $\partial M$, and I am hoping the inequality
$$\lVert Tu \rVert_{L^2(\partial M)} \leq c\lVert u \rVert_{H^1(M)}$$
will hold.
...
3
votes
0
answers
95
views
Strengthening of the local smoothing estimates for the free Laplacian
The classical local-smoothing estimates for the free Laplacian asserts that:
$$\Vert e^{-it\Delta}f\Vert_{L^2((-\infty,+\infty)\,;\,H^{1/2}(B))}\leq C_B\cdot\Vert f\Vert_{L^2}$$
where $B\subset\mathbb{...
- 943
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degree theory for elliptic equations; special solutions
I am interested in using degree theory to examine some semilinear problems.
But instead of just looking for solutions lets assume i am looking for a certain class of solutions; for instance lets ...
- 1,385
1
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1
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1k
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A property on the Green-St Venant strain tensor
Green-St Venant strain tensor is defined by $E(u)={1\over 2}[\nabla u+(\nabla u)^T+(\nabla u)^T\nabla u]$, where $\nabla u$ is the displacement gradient.
Show that
$u\in H^1(\Omega), E(u)\in L^r(\...
- 129
1
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1
answer
209
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Strong maximum principle for the heat equation in non-cylindrical domains
let $u(t,x)$ be a bounded smooth solution of the heat equation $u_t=\Delta u$, $(t,x) \in R \times R^2$, and let $V \subset (R \times R^2)$ be an open connected component of $\{(t,x) \in R \times R^2: ...
3
votes
1
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678
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Is this kernel space of finite dimension ?
Assume that $P \in \Psi^{m}(X)$ (X is a $C^{\infty}$ manifold)is properly supported and has a real principal part p which is homogeneous of degree m.I'm interested in the existence theorem(at least ...
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