All Questions
Tagged with real-analysis ap.analysis-of-pdes
569 questions
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Eigenvalues of elliptic operator analytic with respect to a parameter
I am interested when one can say the eigenvalues of an elliptic operator
are real analytic with respect to a parameter. In particular I have seen many people say the first eigenvalue is analytic but ...
- 1,385
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0
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105
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Positivity of solution of Poisson equation
Let $B$ denote the unit ball centered at the origin in $R^N$ and take $N \ge 3$. Let $( \phi_k(\theta), \lambda_k)$ for $ k \ge 0$ denote the Eigenpairs of $ -\Delta_\theta$ on $S^{N-1}$ which are $L^...
- 19.6k
3
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1
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Distance function is unique nonnegative continuous function on $\mathbb{R}^d$ satisfying following
Suppose $U \subsetneq \mathbb{R}^d$ is open. How do I see that the distance function$$u(x) = \min_{y \in \mathbb{R}^d \setminus U} |x - y|$$is the unique nonnegative continuous function on $\mathbb{R}^...
- 5,169
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1
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Radial Kernel with Bounded Support and Norm of Gradient Bounded by a Dimension-free Constant
I was wondering if it is possible to construct a compactly supported radial kernel function in $\mathbb{R}^d$ such that the norm of the gradient is bounded by some dimension-free constant. That is, ...
CommunityBot
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148
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When a PDE add a Laplacian term
I went to a talk today and the speaker mentioned when you add a Laplacian term to a PDE, the Laplacian will dominate (in what sense?), which I don't quite understand. I know this question is a bit ...
- 39
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308
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Invertible operator
We consider the operator $$T=I + {{{\partial ^2}} \over {\partial {x^2}}}:{H^2}(0,L) \cap H_0^1(0,L) \to {L^2}(0,L)$$
We hope to prove that $T$ is invertible if and only if $L = n\pi $.
and for this ...
- 6,259
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3
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3k
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Dual space of $L^2(\mathbb{R},L^1(0,1))$?
I was wondering what the dual space of $L^2(\mathbb{R},L^1(0,1))$ is? (equipped with Lebesgue measures)
Formally, one would suspect that it is just $L^2(\mathbb{R},L^{\infty}(0,1))$. But this may be a ...
CommunityBot
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1
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130
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Resolvent difference of absolute values!
Let $T$ be a bounded operator. Then, the operators $\left\lvert T \right\rvert:=\sqrt{T^*T}$ and $\left\lvert T^* \right\rvert:=\sqrt{TT^*}$ are well-defined.
Is there a way to write
$$(\left\lvert ...
CommunityBot
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1
answer
102
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Evolution equation invariance of sets
Let $A: D(A) \subset X \rightarrow X$ be a generator of a $C_0-$semigroup and $Z$ be a bounded operator on $X$, then the evolution equation for $u \in C([0,T], \mathbb{R})$
$$\varphi'(t) = A \varphi(t)...
- 16.4k
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226
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degree theory argument in elliptic pde; apparent contradiction
i have a question regarding a degree theory argument and an apparent contradiction. Let me point out that I am a complete novice with degree theory and really i am just pushing some symbols with no ...
- 1,385
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180
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Implicit function theorem for operators
Let $P: (-a,a) \rightarrow \Psi_h^0(\mathbb{R}),$ be a pseudodifferential operator in Weyl quantization with $(-a,a) \ni z \mapsto P(z)$ depending smoothly on this parameter $z$. Note that this ...
CommunityBot
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280
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Helmholtz-Hodge decomposition
I have a question regarding a decomposition of a vector field. So fix $ 1<p<\infty$ and let $ \Omega$ denote a smooth bounded domain in $ R^N$. Now let $ F $ denote a smooth vector field $F:\...
- 1,385
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Trace-norm of integral operator
Let me start by saying that I do appreciate any insight on this. So also if you have a partial result, please share it as a comment or answer.
This is somewhat unrelated to what I normally do, so I ...
- 24.2k
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1
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104
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Operator identity for convergent series
Let $T_i$ and $S_i$ be a sequence of bounded operators such that
$$ \sum_{k,i,j=0}^{\infty} S_j^* T_i^* T_i S_k$$ converges unconditionally in operator norm on some Hilbert space. The limit is then ...
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115
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Does this Sobolev-space like construction have a name?
Take $\Omega \subset \mathbb{R}^n$ arbitrary then define as $X$ the closure of $C^1(\Omega) \cap W^{1,1}(\Omega)$ w.r.t. the norm $f \mapsto \left\lVert f \right\rVert_{\infty} + \left\lVert \nabla f \...
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518
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Interpolation between Schatten classes
I was wondering if there is an analogue to the classical Riesz Thorin theorem for Schatten classes. I suppose the answer is yes, since Schatten classes are so similar to $\ell^p$ spaces for which the ...
2
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2
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232
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Heat equation: impact of the diffusion coefficient on the Harnack constant
Consider the heat equation
$$
u_t - div[a(x,t) \nabla u] =0,\quad (x,t) \in B(r) \times [-r^2, 0] \subset \mathbb R^{d+1}
$$
for a Hölder continuous coefficient $a(x,t)$ satisfying
$$
0<C_o \le a(x,...
CommunityBot
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1
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290
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Any viscosity solution must be the distance function?
Suppose $U \subseteq \mathbb{R}^d$ is open and bounded. Is it possible anybody could supply a simple proof that any viscosity solution of$$\begin{cases} |Du| = 1 & \text{in }U \\ u = 0 & \text{...
- 595
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Direct proof of fact $u \in C(U)$ satisfies $|Du| \ge 1$ in sense of viscosity if and only if property holds
Is it possible anybody could sketch me a direct proof of the fact that $u \in C(U)$ satisfies $|Du| \ge 1$ in the sense of viscosity if and only if the following property holds?
If $V \subseteq U$ is ...
- 349
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views
An argument in the proof of a compactness theorem
In the proof of a compactness theorem involving fractional derivatives in Temam's Navier-Stokes Equations, an argument as the following is made.
Suppose $X_0,X,X_1$ are Hilbert spaces such that
...
CommunityBot
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105
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compactness of sequence of harmonic functions
Let $ \Omega$ denote a smooth bounded domain in $ R^N$ and let $u_m \in C^\infty( \overline{\Omega})$ harmonic functions. We also suppose $ u_m$ is bounded in $L^2(\Omega)$ (uniformly in $m$).
...
- 1,385
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331
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Verifying a claim regarding $H^1$ weak convergence and $L^2$ strong convergence on a surface
I'm reading a paper whose first section discussed $H^1$ maps defined on star-shaped sets, but I got stuck in verifying a claim for quite a while. I'm now thinking the claim is wrong, but it's hard to ...
- 1,350
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99
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Existence of a viscosity solution
Setup
I'm trying to find sufficient conditions for the existence of a viscosity solution to the following PDE,
$$
f(t,s,z) + \partial_sf(t,s,z) \\
+ \sum_{i=1}^{\infty} \left[
\partial_{z_i} f(t,s,z)...
- 5,405
2
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1
answer
116
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Bounding a function with second moments
Let $f(x,y)$ be a non-negative function with $x,y \in \mathbb R^3$ that satisfies
$$
I_1(f) := \iint_{\mathbb R^3 \times \mathbb R^3 } f(x,y) \, dx \ dy < \infty
$$
and
$$
I_2(f) := \iint_{\...
CommunityBot
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1
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95
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Estimating pointwise multiplication conjugated by a Fourier multiplier
I asked this question first on MSE but there was no activity.
Let $m(D)$ be a Fourier multiplier and $f$ a known function. I'm trying to estimate the operator
$$Tu=m^{-1}(D)(f(x)m(D)u)$$
in say $H^s$....
CommunityBot
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128
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determine when $e^{ikx}$ can be boundary value of a holomorphic function
Assume that $\Gamma=\{x+if(x): x\in \mathbb{R}\}$ is a graph, separating $\mathbb{C}$ into two connected components. Let's denote the one below $\Gamma$ by $\Omega$.
My question is, for what curves $...
- 24.2k
8
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2
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634
views
Existence of a uniformly continuous function $g$ on $\mathbb{R}$ where $f = g$ a.e.?
Suppose $f \in L^\infty(\mathbb{R})$, $f_h(x) = f(x + h)$, and$$\lim_{h \to 0} \|f_h - f\|_\infty = 0.$$Does there exist a uniformly continuous function $g$ on $\mathbb{R}$ such that $f = g$ almost ...
- 3,709
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Given a fixed convex domain $\Omega$ in 3D, for what value $c$ the function $f(c) := \int_{\partial \Omega} |x-c| d \sigma_x$ gets its minimum?
Let $\Omega$ be a bounded smooth convex domain in $\mathbb{R}^3$, then consider the following minimization problem:
$$\inf_{c \in \overline{\Omega}} f(c), \quad f(c) := \int_{\partial \Omega} |x-c| ...
- 1,350
3
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0
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588
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Time-dependent Sobolev spaces
Given the Sobolev space $H^1((a,b);H^2(\mathbb{R}))$ and a function $g$ in that space. Consider now another function $f \in C_c^{\infty}((a,b) \times \mathbb{R}).$ Then
for almost any $t \in (a,b)$ we ...
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1
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187
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Free quantum evolution operator on Sobolev space
I am not a mathematician, but would like really like to get some confirmation on the things I am doing here.
Let $-\Delta: H^2(\mathbb{R}) \subset L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ then ...
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394
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Fixed radius mean value property implies harmonicity?
Let $f$ be a continuous real-valued function on $\mathbb{R}^n$. It is well known that the following are equivalent:
$f$ is harmonic.
$f$ satisfies the ball mean value property
$$
f(x)=\frac{1}{|B(x,r)...
CommunityBot
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117
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The eigenfunction of modified $1$-laplace equation?
Let $\Omega\subset \mathbb R^2$ be open bounded with smooth boundary. It is well known that the laplace equation
$$
-\Delta u=0
$$
has a set of eigenvalues $0<\lambda_1<\lambda_2\leq\lambda_3<...
- 679
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votes
1
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160
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Global Poincaré type estimate
For simplicity let us assume we are considering $\mathbb{R}^3$. Let us define the weighted Sobolev norm $\| u \|^2_{L^2_{\alpha}}= \int_{\mathbb{R}^3} |u|^2 \langle x\rangle^{\alpha}$ where $\langle x ...
- 4,143
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2
answers
953
views
Differentiability of Nemytskii operator on Sobolev space
I am trying to consider hypothesis on $g$ such that the operator
$$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$
is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...
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The jump set of $SBV$ function for different value of parameter in image denoising problem
The classical Mumford-Shah image denoisng problem study the minimizer of the following functional, for each $\alpha>0$ where $\Omega\subset \mathbb R^2$ is open bounded with sommth boundary,
$$
u_\...
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If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$
Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$.
Given then $f\in L^1(\Bbb R^n)$; by density there exists $\{f_j\}_j\...
CommunityBot
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3
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2k
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Gross's log Sobolev inequality proof with variational calculus?
For $f\in C^{1}(\mathbb{R}^{n})$, Gross's logarithmic Sobolev inequality says that
$$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left(\int f^{2}\,d\mu\right)\leq \frac{2}{c}\int |\nabla f|^{2}...
- 27.5k
2
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86
views
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 $...
CommunityBot
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1
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Step 2 of The Strichartz's Estimates in Cazenave's Book
My question is from Cazenave's book "Semilinear Schrödinger Equation", page 35. I am stuck with Step 2 of the Strichartz's estimates.
The book says that $||\Phi_f(t)||_{L^2}^2=\left(\int_0^t \...
- 39.1k
4
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1
answer
1k
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Density argument with Schwartz functions?
I was wondering whether the Schwartz functions are also dense in
$$\{f \in L^2(\mathbb{R}^n); \int_{\mathbb{R}^n} |x|^2 |f(x)|^2 dx + \int_{\mathbb{R}^n}|\xi|^2 |\hat{f}(\xi)|^2 d \xi < \infty\}$$
...
- 24.2k
1
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0
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192
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The decay rate of the spectrum of the Gaussian kernel on compact manifolds
It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...
- 617
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1
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2k
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A continuity/bootstrap argument
I am trying to understand how one can prove the following assertion using a continuity argument:
Let $0<\epsilon<\epsilon_0$. Let $I=[t_0,R]$ be a compact interval. Suppose that $S:I\to [0,\...
- 5,169
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votes
1
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374
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Simplify proof for rapidly decaying functions
I want to show the following theorem in a lecture:
Let $F \in C^{\infty}(\mathbb{C}^{k}, \mathbb{C})$ such that $F(0)=0.$
Let $G: \mathbb{R}^n \rightarrow \mathbb{C}^{k}$, $x \mapsto (f_1(x),..,f_k(...
- 2,243
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0
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131
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Well-definedness on $C_{0}^{\infty}(\mathbb{R}^{n})$
Let $T$ be a Calderon-Zygmund operator associated to a Calderon-Zygmund kernel $K\in CZK_{\alpha}$ of order $\alpha>0$ and $b\in BMO(\mathbb{R}^{n})$. Then for $f\in C_{0}^{\infty}(\mathbb{R}^{n})$ ...
- 355
1
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1
answer
129
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$L^p$-bounding inequality [closed]
Do we have that$$\|Du\|_{L^{2p}} \le C\|u\|_{L^\infty}^{1\over2} \|D^2u\|_{L^p}^{1\over2}$$for $1 \le p < \infty$ and all $u \in C_c^\infty(U)$? Here, $U$ denotes an open subset of $\mathbb{R}^n$.
- 16.2k
9
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2
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553
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Asymptotic behavior of Sturm-Liouville eigenvalues
I have two questions.
Consider the operator $Av = -v'' + a(x)v$ on $I = (0, L)$, with zero Dirichlet condition and $a \in C([0, L])$.
Let $(\lambda_n)$ denote the sequence of eigenvalues of $A$....
- 205
1
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0
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92
views
Perturbation in Besov space
$\|f\|_{B^{0}_{p,p}}=(\sum_{j\geq -1} \|\Delta_j f\|_p^p)^{1/p}$ is the Besov norm of $f$.
Here the Fourier transform of $\Delta_jf~(j\geq 0)$ is $\psi(2^{-j}\xi)\hat{f}(\xi)$ and $\psi$ is a smooth ...
- 515
2
votes
1
answer
301
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Simplicity of eigenvalues
Consider the Sturm-Liouville operator$$Au = -(pu')' + qu \text{ on }I = (0, 1),$$where $p \in C([0, 1])$, $p \ge \alpha > 0$ on $I$, and $q \in C([0, 1])$. No further assumptions are made; in ...
- 52.3k
5
votes
1
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133
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If $u \in H^1(U)$, then $Du = 0$ almost everywhere on the set $\{u = 0\}$, auxiliary result
Let $\phi$ be a smooth, bounded and nondecreasing function, such that $\phi'$ is bounded and $\phi(z) = z$ if $|z| \le 1$. Set$$u^\epsilon(x) := \epsilon \phi(u/\epsilon).$$Do we necessarily have that$...
- 78
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1
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843
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$C^{\infty}_{loc}$-convergence - right definition
Let $\Omega \subset \mathbb{R}^{n}$ be some open set. Let $f_{n},f\in C^{\infty}(\Omega)$. My question is: What does the following phrase mean? $f_{n}$ converges to $f$ in $C^{\infty}_{loc}(\Omega)$. ...
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