All Questions
546 questions with no upvoted or accepted answers
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Existence of $C^{2, \alpha}$ solution to $a^{ij}(x,u,Du)D_{ij}u+b(x,u,Du)=0$ using the Leray–Schauder theorem in "Elliptic PDE" of Q. Han & F. Lin
In this part of the book "Elliptic PDE" of Qing Han & Fanghua Lin, the Leray–Schauder existence theorem is applied to prove the existence of $C^{2, \alpha}(\bar{\Omega})$ solution.
For $\...
- 809
1
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0
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131
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Results on the eigenspace of weighted elliptic eigenvalue problems
I am considering the following eigenvalue problem in $\Omega=\mathbb{R}^n_+$
$$-\operatorname{div}(a(x)\nabla \varphi) = \lambda a(x)w(x)\varphi$$
where the weights $a>0$ and $w\in L^{\infty}$ (and ...
- 537
1
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0
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299
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Is there a generalization of the Agmon-Douglis-Nirenberg regularity theorem for elliptic equations to domains with corners?
The Agmon-Douglis-Nirenberg theorem(s) state(s) that whenever $f\in W^{m,p}(\Omega)$ where $\Omega$ is a bounded open set of class $C^{m+2}$, then there is a unique solution $u\in W^{m+2,p}(\Omega)$ ...
- 597
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103
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Regularity results for non uniform elliptic equation
I have seen some regularity result for ellptic PDE but all of them consist of uniform elliptic one. For instance,
$$\nabla \cdot (\gamma(x) \nabla u)=F \text{ in } \Omega\qquad u=\phi \text{ on }\...
- 309
1
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0
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251
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Regularity of a Fokker-Planck PDE with unbounded coefficient
Let $A$ be a positive definite symmetric matrix, let $b\in C^1(\mathbb R^d\!\times\!(0,\infty))\cap C(\mathbb R^d\!\times\
- 311
1
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0
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72
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Initial-boundary value problem for transport equation with $W^{1,p}$ velocity
Let us consider $v:\mathbb R_+ \times \mathbb R \to \mathbb R_+$ such that $v \in L^1(0,\infty, W^{1,p}(\mathbb R))$ and the transport equation
$$ \begin{cases}
u_t + v(t,x) u_x = 0 \qquad & (...
- 61
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105
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Derivative and Green function of Fractional Laplacian in a bounded domain: $(-\Delta)^s\nabla_x G(\bar x,z) = 0 \text{ in } \Omega $?
Let $G$ be the Green function of the Fractional Laplacian $(-\Delta)^s$ in a domain $\Omega$ (which is known explicitly for the special case of the ball: link). Let $\bar x \in \Omega$ be fixed. Does ...
- 303
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0
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94
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Obtaining identity from Pokhozhaev formula
From the classical Pokhozhaev formula, how can I obtain that the following identity holds for $u,v \in C^2(\bar \Omega)$?
$$
\int_\Omega (\Delta u(x,\nabla v) + \Delta v(x,\nabla v)) dx = \int_{\...
- 217
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224
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Is this mixed-Sobolev interpolation inequality valid?
By $L^p_T\dot H^s$ I mean the space $L^p([0, T] \to \dot H^s)$ where as usual
$$\|u\|_{\dot H^s}^2 = \int_{\mathbb R^d} |\hat u(\xi)|^2 |\xi|^{2s} ~d\xi.$$
Of course, we can interpolate in the time ...
- 708
1
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0
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258
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Cut-off function and fractional Laplacian
Is there a smooth function $u$ such that $u = 1$ in $B_r(0)$, $u=0$ in $\mathbb R^N \setminus B_{2r}(0)$, and
$$
|\nabla u| \le Cr^{-1}, \quad |\Delta u |\le Cr^{-2}, \quad |(-\Delta)^s| u \le Cr^{-2s}...
- 99
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36
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Existence and uniqueness for fractional parabolic equation with transport term
Let us consider the problem
\begin{equation}
\begin{cases}
u_t+(-\Delta)^{\sigma}u+\mathrm{div}(a(t,x)u) = 0 & \text{in } \mathbb{R}^n \times [0, T) \\
u(x,0)=u_0(x) & \text{in } \...
- 99
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0
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144
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Liouville theorem for elliptic equation with advection term
How can one prove that any $L^2$ solution of
$$ - \Delta \phi(x) + a(x) \cdot \nabla \phi(x)=0 \qquad \mbox{in } \mathbb R^N $$
is zero if $a(x)$ is a divergence-free vector field such that
$\int |\...
- 99
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0
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130
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Fractional Sobolev embedding theorem
Let $\psi \in C^\infty_c(\mathbb R^N)$ be a test function with support iN $B(0,R)$. Is it true that the following inequality holds
$$\int_{B(0,R)} \psi^2 u^{\frac{4}{1+\beta}} dx \le R^{1+\beta} \int_{...
- 99
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122
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Series and solution of $-\Delta u + \lambda u = f(x)$
Consider a bounded smooth set $\Omega \subset \mathbb R^n$ (for example, we can take a ball). Can we write down the solution of
\begin{align*}
-\Delta u(x) + \lambda u(x) &= f(x), & x \in \...
- 131
1
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0
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42
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On the boundary integral of Neumann eigenfunctions
Let $v$ be an eigenfunction corresponding to the first nonzero Neumann Laplacian eigenvalue on a domain $\Omega \subset \mathbb{R}^2$. By definition, we know that $\int_{\Omega} v \, dx=0$. If $\Omega$...
- 1,350
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40
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Various definitions of weak solutions and continuity property
For example, consider the transport equation
$$
\partial_tu+b\cdot\partial_xu=0, u(0,x)=u_0(x),
$$
where $t\ge0,x\in\mathbb{R}^3$, $b\in W^{1,1}_{\rm{loc}}([0,\infty)\times\mathbb{R}^3)$.
It seems ...
- 71
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81
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Compact imbedding for weight space
We begin with some definitions. Let $\gamma \geqslant 1,\,p \in \left[ {1,\infty } \right)$, we define
$$L_\gamma ^p\left( {0,1} \right) = \left\{ {v:\left( {0,1} \right) \to \mathbb{R}:{{\left\| v \...
- 183
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119
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Can $H_{\mathrm{rad}}^s(\mathbb{R}^n)$ compactly embedded in $L^{\sigma}(\mathbb{R}^n)$?
$\DeclareMathOperator\rad{rad}$
Can $H_{\mathrm{rad}}^s(\mathbb{R}^n)$ be compactly embedded in $L^{\sigma}(\mathbb{R}^n)$?
In $H_{\rad}^1(\mathbb{R}^3)$, by Struass estimate $|f(x)| \lesssim |x|^{-1} ...
- 429
1
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0
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42
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Energy estimate for $\theta_t + H(\theta)_x = 0$ in $t>0, x >0$?
Consider the IBVP for $$\theta_t + H(\theta)_x = 0, \qquad t>0, \ x>0$$ with $$H(\theta) = \frac{1}{\pi} \text{pv}\int_{0}^\infty \frac{\theta(y)}{y-x} dy$$
with Dirichlet boundary conditions. ...
- 161
1
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0
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74
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Fourier transform of a Sobolev function dependent on a "parameter"
Let $u\in\mathcal{S}(\mathbb{R}^n)$, let $V\in W^{1,1}_\text{loc}(\mathbb{R}^n\times\mathbb{R}^+)$, such that
$$ V(x,0)=u(x),\quad V(x,\cdot)\in C^0([0,\infty)),\quad\forall x\in\mathbb{R}^n,$$
and ...
- 339
1
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0
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213
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Fractional Laplacian extension problem and uniqueness question
I am studying the article "An Extension Problem Related to the Fractional Laplacian" by L. Caffarelli and L. Silvestre. Consider the following problem:
$$ \Delta_xu+\frac{a}{y}u_y+u_{yy}=0, $...
- 339
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88
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Reference request: PDE of the form $(\Delta - |u|^2)f = F(u)$
I am interested in equations of the form
$$(\Delta -|u|^2)f = F(u)$$
where $F$ depends on $u$ and preferably on its derivative, too. $u$ is supposed to be given and $f$ the unknown. More precisely I ...
- 469
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0
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66
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Well-posedness of hyperbolic system with constant coefficients in finite domains
I'm studying the PDE
$$
\frac{\partial u}{\partial t} + A_x\frac{\partial u}{\partial x} + A_y\frac{\partial u}{\partial y} + A_z\frac{\partial u}{\partial z} = 0
$$
with $A_x, A_y, A_z$ being ...
- 11
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0
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84
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A Riemann Hilbert problem on the unit square
Let $p_0=0$, $p_1=1$, $p_2=1+i$ and $p_3=i$ be the four vertices of a square $Q$ on the complex plane $\mathbb C$.
Let $f \in C^{\infty}_c((0,1))$ and consider the following Riemann-Hilbert problem on ...
- 4,143
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353
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Eigenvalues of convolution matrices
Let $h: \mathbb{R}\to \mathbb{R}$ be a smooth function. Fix $0\leq s_1\leq \cdots \leq s_m\leq 1$ and $0\leq t_1\leq \cdots \leq t_n\leq 1$. Construct $A\in \mathbb{R}^{m\times n}$ by letting $A_{i,j}:...
- 315
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126
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Does my functional satisfy the Palais Smale condition?
Consider the functional
$$ I(u)=\frac{1}{2} \int_\Omega |\nabla u|^2\ dx + \frac{1}{4} \int_\Omega (1-|u|^2)^2 \ dx - \frac{c}{2} \int_\Omega \langle i\partial_1 u , u\rangle ,$$
where $u:\mathbb{R}^2 ...
- 383
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126
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Continuity of Helmholtz-Hodge projection in $H^1(\Omega)$
Let $\Omega \subset \mathbb{R}^d$ (for simplicity $d = 2$ or $3$) be a bounded Lipschitz domain. For any vector-valued function $\mathbf{f} \in \mathbf{L}^2(\Omega):= \left ( L^2(\Omega) \right )^d$, ...
- 23
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0
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148
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Spectrum of Laplacian-like operator
Let $\kappa_1, \kappa_2>0$ be fixed.
Consider the unbounded operator $A: D(A) \rightarrow L^2(-1,1)\times\mathbb{R}$ defined by
$$
A\begin{bmatrix} y \\ h \end{bmatrix} = \begin{bmatrix} \...
- 309
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0
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64
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Bound on number of linearly independent eigenvectors of adjoint of composition operator
Fix $N>1$. Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that the composition operator via
$$
\begin{aligned}
C_f:C^{\infty}(\mathbb{R},\mathbb{R}) &\rightarrow C^{\infty}(\mathbb{R},\...
- 5,405
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0
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102
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Commutator estimates for $-(-\Delta)^s$, with $s \in (1,2)$
I'm currently trying to work with the non-local operator given by
$$
(-\Delta)^{\frac{s}{2}}f(x)= c_s\text{P.V} \int_{-\infty}^\infty \frac{-f(x+y)-f(x-y)+2f(x)}{|y|^{1+s}} dy,
$$
where $f :\mathbb ...
- 446
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0
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45
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Decomposition of the space of Radon measures with respect fractional harmonic capacity?
It is well know that there is a generalization of Lebesgue decomposition theorem in the following way:
Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely ...
- 371
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169
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A question about Stroock's notes on the Weyl lemma
On p.4 of these notes, D. Stroock gives a quick and efficient construction of the Markov transition functions of a certain diffusion. The idea of his construction (on page 4) is to 'freeze' the ...
- 1,461
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136
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Topology on function spaces for pointwise convergence
Suppose I have some collection of maps $T_\lambda: C^\infty(\Omega)\rightarrow C^\infty(\Omega)$ which are linear and parameterized by a parameter $\lambda >0$. (Perhaps more generally take $C^\...
- 11
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88
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Explanation for the energy method used here
I am reading a paper where the authors prove
$$ \frac{d}{dt} \Vert f \Vert_{H^4} (t)\leq C\Vert f \Vert^5_{H^4}(t) $$
Where $f=f(x,t)$ and $H^4$ stands for the usual Sobolev space. Using Gronwall's ...
- 383
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0
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166
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Solution to Heat Equation By Projection
Let $X\subseteq C(\mathbb{R}\times [0,\infty))$ be the collection of all solutions to the IV heat equation
$$
\partial_t u(t,x) = \partial^2 u(t,x) \qquad u(0,x)=p(0,x),
$$
for some fixed $p\in C^2(\...
- 5,405
1
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0
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80
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One dimensional periodic travelling waves to some pde
Travelling wave equation on one dimension to Gross Pitaeavkii equation is $$ \phi '' +ic\phi'+\phi (1-|\phi|^2)=0\qquad (1) $$ where $c\in (0,\sqrt{2})$ and $ \phi$ is a complex valued function. I am ...
- 383
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110
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Trace and second-order inverse trace on space with Gibbs measure
Consider $(t, x)\in [0,T]\times (\mathbb{R}^d,d\mu)$, where the measure $d\mu(x)=K^{-1}\exp(-U(x))dx$ is a reasonable Gibbs measure (it satisfies a Poincaré or log-Sobolev inequality. One can, for ...
- 73
1
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0
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108
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Extension of a compactly supported pseudo-differential operator
Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ and $P \in \Psi^{m}(\Omega)$ a compactly supported pseudo-differential operator, that is, the kernel of $P$, is compact. Is it true that $P$ extends ...
- 509
1
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0
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177
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A consequence of De Giorgi oscillation lemma
The following lemma is true (see DeGiorgi oscillation lemma)
Let $u$ be a subsolution of $$\mathrm{div}(A(x)\nabla u) = 0,$$
where $A$ is bounded, measurable and uniformly elliptic ($C^{-1}\...
- 839
1
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0
answers
52
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Asymptotically periodic potentials
Who came up with the idea of solving elliptic equations with periodic potentials and from there solving elliptic equations with asymptotically periodic potentials?
- 69
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0
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48
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Integrability condition on function determining PDE domain
I'm currently looking through the following paper which examines some dynamics of the Airy$_2$ process: https://arxiv.org/pdf/1106.2717.pdf
On page 2, there appears a PDE of the form
$\partial_t u +...
- 123
1
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0
answers
62
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Wellposedness of semilinear wave equation with discontinuous source
Where can I find existence and uniqueness results for semilinear wave equations with discontinuous, i.e.
$$\partial^2_{tt} u - \Delta u = f(u), \quad t >0, \ x \in \Omega$$ where $f$ is ...
- 277
1
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0
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75
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Derivation of the vortex filament equation from Euler equation
How can the vortex filament equation
$$\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 \...
- 277
1
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0
answers
87
views
Global solution of nonlinear Schrödinger equation via blow-up argument
Set $u_0\in H^1 (\mathbb{R} ^N)$ and $1<\alpha < \frac{N+2}{N-2}$.
I want to show that there exists $\varepsilon > 0$ s.t. if $\Vert u _0 \Vert _ {H^1} < \varepsilon$,
then there is ...
- 383
1
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0
answers
103
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Choosing the weight in a particular definition of Besov spaces
Following Giovanni Leoni's excellent book (or the Wikipedia article) one possible way to define the Besov spaces $B^{s,p,\theta}(\mathbb R ^d)$, with $s\in(0,1)$ the fractional "order of derivative" ...
- 5,380
1
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0
answers
78
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Inequality for fractional power norm (sectorial operators)
How could we prove following inequality:
$\int\limits_{0}^{l} u^{3}(x) dx \leq \sqrt{l} \cdot|| u||_{\frac{1}{2}}^{3}$
where
$ || u ||_{\frac{1}{2}} = ||A^{\frac{1}{2}}(u)||_{L^{2}} + || u ||_{L^{...
- 31
1
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0
answers
92
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Alberti rank-one theorem and reduction of the study of BV function to the two-dimensional case
By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At ...
user123457
1
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0
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184
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One question about Schrodinger Semigroups-(B. Simon)
This question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3).
On the Theorem C.3.4(subsolution estimate) of the paper, it says that: Let $Hu=Eu$ and $u\in L^...
- 1,915
1
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0
answers
177
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Singular integral of the composition of the Hilbert transform and fractional Laplacian
Given $0<s<1$, we can define the Fractional Laplacian by
$$\Lambda^{-s}f(x):=(-\Delta)^{-s/2}(x)=\int_{-\infty}^{+\infty}|x-y|^{-1+s}f(y)dy$$
or by means of Fourier transform as $$\widehat{\...
- 111
1
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0
answers
74
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Elliptic pde L^p theory via adjoint theory
Let $ T:X \rightarrow Y$ denote some linear operator and suppose we know its one to one (here $X$ and $Y$ are Banach spaces). I believe their is results that say $Ker(T^*)= (R(T))^\perp$ (where ...
- 1,385