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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variation of the obstacle in the obstacle problem
Suppose $D \subset \Bbb C$ with smooth boundary. Let $f \in C^{1,1}(D)$. Let $\varphi$ be the supremum of all members in the set
$$\lbrace g \in C^{\infty}(\overline{D})| g \ is \ subharmonic \ and \...
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weak derivative and continuous function
Let $\Omega \subset \mathbb{R}^n$ be a compact smooth hypersurface. Suppose $\varphi \in C_c^\infty(0,T; H^1(\Omega))$ is a $H^1(\Omega)$-valued test function (so $\varphi(t) \in H^1(\Omega)$ for each ...
CommunityBot
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Fourier transform of function on compact set and Sobolev norm equivalence
Hi all. My question on M.SE is unanswered (https://math.stackexchange.com/questions/254970/fourier-transform-of-function-defined-on-subset-of-mathbbrn) so I want to post it here. I changed it slightly....
CommunityBot
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Conditions on a unit vector field to be the Gauss map of some surface immersed in R^3?
Let $U$ be a bounded domain in $R^2$ and let $n : U \to S^2$. Which (necessary/sufficient) conditions must $n$ satisfy in order that there exist an immersion $f : U \to R^3$ such that $n(x)$ is the ...
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Hormander's bracket condition for the adjoint of an operator
Let $X_0, X_1, \dots, X_k$ be smooth vector fields over ${\mathbb R}^n$, and let us consider the operator
$$
L = \sum_{i=1}^k X_i^2 + X_0~.
$$
Here, I assume that Hörmander's bracket condition is ...
CommunityBot
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C^\infty versus semiclassical wavefront sets
Zworski states that if $u$ is a compactly supported distribution, independent of the semiclassical parameter $h$, then the relationship between the $C^\infty$ and semiclassical wavefront sets of $u$ ...
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Please recommend some classical books or articles on the compressible Euler equations !
Please recommend some classical books or articles on the local well-posedness result of compressible Euler equations ! The main aim is that I want to learn some basic methods and techniques about the ...
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141
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Lagrangean uniqueness versus Eulerian uniqueness
(1) Lagrangean description. Let us consider a $N\times N$ system of autonomous ODE:
$$
\dot x=a(x),\quad \mathbb R\ni t\mapsto x(t)\in \mathbb R^N,\quad a:\mathbb R^N\rightarrow \mathbb R^N.
$$
...
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Willmore minimizers for genus $\geq 2$
For an immersed closed surface $f: \Sigma \rightarrow \mathbb R^3$ the Willmore functional is defined as
$$
\cal W(f) = \int _{\Sigma} \frac{1}{4} |\vec H|^2 d \mu_g,
$$
where $\vec H$ is the mean ...
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damped wave equation
For $t>0$, $x$ in a compact Riemannian manifold $(M,g)$, and $a\in C^\infty(M)$, $a\geq0$, $(\partial_t^2+a\partial_t-\Delta_g)u=0$ is called the damped wave equation.
My question is...why is the ...
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Divergence form Elliptic PDE Removable Singularity/Regularity Question
Idea
Given a $W^{1,2}$ solution to a linear divergence form uniformly elliptic pde with bounded coefficients, standard De Giorgi-Nash-Moser theory tells us that the solution is infact (Holder) ...
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Does Hölder continuity imply smoothness for the CMC equation: $u:D^2\rightarrow\mathbb{R}^n$, $\Delta u = 2H\partial_xu\times\partial_yu$, $H$ constant?
Context: I am currently reading through the freely available lecture notes from Tristan Riviere (here) on the applicability of integration by compensation in the analysis of various geometrically ...
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fixed point arguments in PDE
I was curious whether anyone knows of some examples in PDE where
a standard fixed point argument fails to show the existence of a solution but one can apply
one of the more advanced fixed point ...
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2
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Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform
Question: Suppose $a(x,y)\in C^\infty([0,1]\times [0,1])$ and suppose
$$\sup_{\lambda>1} \bigg|\lambda\int_0^1 e^{\lambda x} a(x,1/\lambda)dx\bigg|<\infty.$$
Is $a(x,0)=0$, $\forall x\in[0,1]$?
...
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Showing a coercivity condition for this bilinear form
Suppose $\Omega \subset \mathbb{R}^n$ is a compact domain. Let $f$ and $J$ (and also $\frac 1J$) be $C^1$ functions on $\Omega$. Consider the bilinear form $a:H^1(\Omega) \times H^1(\Omega) \to \...
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Does this PDE has a general solution? [closed]
$$K\frac{\partial }{{\partial x}}(h\frac{{\partial h}}{{\partial x}}) = \mu \frac{{\partial h}}{{\partial t}}$$
K and u are constants.
If no,how to get a asymptotic solution?ie,linearize
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Change in neumann boundary conditions through coordinate transformation of elliptic PDE, weak formulation
The standard weak formulation of the Neumann problem for the Poisson equation
is to find $u \in H^1 ( \Omega)$ such that for every $v \in H^1 ( \Omega)$:
$$ \int_{\Omega} \nabla u \nabla v d x = \...
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0
answers
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two polynomial equations
Let $f:\mathbb R^2\rightarrow\mathbb R$ be a smooth function such that for every point $(x,y)\in\mathbb R^2$ the system
$$f_{11}+2tf_{12}+t^2f_{22}=0$$
$$f_{111}+3tf_{112}+3t^2f_{122}+t^3f_{222}=0$$
...
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Subharmonic envelope
I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me know....
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Green functions on Riemann surfaces
Let $(M,g)$ be a compact Rieamnnian surface without boundary and $\Delta_g$ be the Lapalce operator. We note $\lambda_i$ and $\phi_i$ the eigenvalues and eigenunctions of $\Delta_g$. Let also $G_g$ ...
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Well-posedness of Euler-Poisson system for semiconductors
Is there anyone can recommend me some important literature references about the well-posedness of both Cauchy problem and initial boundary value problem of Euler-Poisson system for semiconductors? ...
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Boundary regularity of quasiconformal homeomorphisms of the unit disk ?
Hello, I asked this question before, but didn't get any response, so I took the liberty of asking once again , with slightly modified version of the question:
Consider an orientation-preserving ...
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Hopf Boundary Point Lemma
The Hopf Boundary Point Lemma
http://en.wikipedia.org/wiki/Hopf_lemma
is a result for the unit normal vector field and the normal derivative.
Is it true if one considers arbitrary directional ...
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Weak divergence implies weak differentiability of components?
Suppose $\Omega$ is an open set in $\Bbb{R}^N$ and $\sigma : \Omega \to \Bbb{R}^N$ is a field with all components belonging to $L^2(\Omega)$.
We say that $\sigma$ has weak divergence if there exists ...
- 16.2k
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hitting probability for integrated Ornstein-Uhlenbeck process
Consider an Ornstein-Uhlenbeck position process:
$dV_t=dB_t-\lambda V_tdt$
$dX_t=V_tdt$
where $B_t,V_t,X_t$ are all in $R^d$ with $d\geq 3$. Let $X_0\neq0$, $V_0=0$ .
Let $r>0$ and $S_r$ be the ...
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2
answers
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Easy question on Sobolev spaces
I understand that this question would be trivial for experts, sorry for that, I just need to clarify things.
So let $S(\mathbb{R}^n)$ denote the Schwartz space on $\mathbb{R}^n$ and $W_p$, $W_q$ are ...
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About the boundedness of the derivative of a function which is in a special function space.
If $f \in C^1 ([0,T] , L^2) \cap C^0 ([0,T] , W^{1,2} )$, $f (t,x) : bounded\; on \; [0,T] \times \Bbb R^n $ then how can I conclude that
$$ \left \| \frac{\partial f}{\partial t} \right \|_{L^\...
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Fourier transform and spectrum of PDOs in $L^p$
Let $K$ be a compact subset in $\mathbb{R}^n$ with $m(K)=0$, Suppose $supp\hat{u}\subset K$ for some $u\in L^p$,where $2\leq p\leq \frac{2n}{n-1}$,can we get $u\equiv 0$ ?
Motivation: If $K$ is a ...
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Variational problems whose lagrangian density depends on derivatives higher than 1.
The usual theory of calculus of variations, as far as I know, is concerned with lagrangian densities which depend on the function and its gradient, namely we try to minimise $\int L(Dw,w,x) dx$. ...
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The conormal derivative of a function
Hi!
I was wondering about the definition of the conormal derivative of a function
$u$ which is given on a domain $\Omega$. It is known that if $-\Delta u = f$, considered
as functionals on $H^1_0(\...
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Elliptic regularity on bad domain
Let $\Omega \subset R^n$ be an open bounded set. Consider the Dirichlet problem
$$
-\triangle u = 0, x \in \Omega, \quad u = \varphi, x \in \partial \Omega.
$$
If $\varphi$ is a continuous function, ...
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Interpretation of a parameter in forming a pseudodifferential operator
In Zworski's Semiclassical Analysis, he defines the following method of quantization: for a symbol $a = a(x,\xi) \in \mathscr{S}(\mathbb{R}^{2n})$ and $u \in \mathscr{S}(\mathbb{R}^n)$,
$$ Op_t(a)u(x)...
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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 ...
- 4,899
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Solvability of a nonlinear elliptic equation
Hi, let $\Omega \subset R^3$ be a bounded smooth domain, consider the elliptic equation
\begin{equation}
-\triangle u + u^2div u = f, \quad x \in \Omega, \quad u\big|_{\partial \Omega} = 0.
\end{...
CommunityBot
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Reference Request: Schauder theory for fourth-order parabolic equations
I am looking for a treatment of fourth order parabolic equations in Holder spaces. More precisely fourth order analogues of Theorems 5.1, 5.2, and 10.1 in Chapter IV of Linear and quasilinear ...
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A question on Schwartz distributions
I have a question on the tempered distributions, namely, continous functionals on Schwartz class endowed with the weak* topology. Is is a Barreled space, say, a space whose convex, balanced, ...
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Solving PDE with Cauchy - Kowalewski Theorem
Hallo,
I have the following PDE that I am trying to solve via the Cauchy-Kowalewski Theorem. But I have no idea how to do it or if its possible. Maybe one of you has an idea. Here is the problem: Let ...
- 108k
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1
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Green's function for a certain elliptic equations with rough coefficients
We know the laplacean operator has a Green function which is smooth away from the boundary. Now, consider a linear operator of the form $\partial_i(a^{ij} \partial_j u)$.We can prove that this ...
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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\...
CommunityBot
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On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis
Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. Then my question is :
...
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Longtime behaviour of the periodic KdV equation
I was wondering if anyone could give a heuristic (i.e. preferably non-technical) explanation of what is the expected longtime behavior of the periodic KdV equation.
Recall the standard KdV equation ...
- 114k
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Rellich-Kondrachov compactness theorem in arbitrary smooth metric measure spaces
Consider a smooth metric measure space in which the integral of a gradient is meaningful. For example in the sense of upper gradients of Heinonen, or on a riemannian manifold with the associated ...
- 1,881
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Ways to decompose a torus for finite element method so that each cell contains a complete revolution of the major radius
I've got a finite element problem involving paths around the interior of a torus. For this particular problem I think I could make things more computationally efficient if each cell in the mesh made ...
- 101
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Discrete Sobolev space of $R^n$ valued maps
Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say
$f:\Omega \...
- 3,388
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1
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Almost analytic continuation
Let $f\in S^{\alpha}$ for some $\alpha \in \mathbb{R}$(which means that f is smooth and satisfies $|D^{\beta}f|\leq C(1+|x|)^{\alpha-\beta}$),a function $\tilde{f}$ on $\mathbb{C}$ is called an almost ...
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A special Integral Kernel
Does there exist either one / general class of non-negative definite , symmetric Integral Kernel map satisfying the following properties ??
$f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$
$K:L^2(\...
- 16.2k
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votes
2
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Decay rate of nonlocal differential operator?
Hi, Moers.
Let $m(\xi) \in S^0$, that is,
$$
|D^\alpha m(\xi)| \leq C<\xi>^{-|\alpha|}, \quad \forall \xi \in R^n.
$$
It's well known that $m(D)$ is bounded in $L^p$ for $1 < p < \infty$.
...
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2
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667
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Implications of a hypothetical blow-up of Navier-Stokes for the mathematical model
Let us suppose that there exists a (initially smooth) solution of NSE that blows up in finite time. Then, in particular, the corresponding velocity field becomes unbounded as time progresses. Which ...
- 52.3k
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votes
1
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2k
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Is there a Seiberg-Witten version of Donaldson-Thomas theory?
Donaldson invariants are a count of instantons (the solutions to a particular elliptic PDE) on 4-manifolds. One thing which makes the theory difficult is a lack of compactness for the moduli spaces of ...
- 399
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1
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418
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Frobenius theorem with lesser regularity
Clearly, if one is given a $C^1$ sub-bundle $V$ of the tangent space of a smooth manifold $M$, wheather $V$ comes from a $C^2$ foliation of the manifold is decided by the conditions of the Frobenius ...
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