All Questions
Tagged with fa.functional-analysis real-analysis
1,447 questions
1
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56
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Moduli of continuity and Wasserstein differentiability of functions between measures
Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability ...
- 11
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0
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79
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Conditions on triangle inequality for integral kernel
Consider $\int_RK(x,y)f(y)dy$, where $K(x,y) \in M_+(R^2)$.
Let $L(t,s)$ be an iterated rearrangement of $K$. Let also $$
A(t,v)=\int_0^{1/v}L(1/t,s)ds,
$$
which is decreasing with $v$ and ...
- 343
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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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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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203
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Construction of weight function to satisfy condition on given functional
Consider the following function :
$$F(z) = \omega(z){\sin^2\left(\frac{c\Gamma(z)}{z}\right)}$$
Here, $\omega(z)$ is a weight we are going to consider
The following two conditions should meet for $\...
- 375
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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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40
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Example of periodic semidifferentiable function without absolutely convergent Fourier series
Is there an example of a periodic continuous function that is semidifferentiable (i.e the left derivative and the right derivative exist at each point), but
with a non-absolutely convergent Fourier ...
- 1,035
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0
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196
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Asymptotic of a functional as $x\rightarrow \infty$
Consider the following functional :
$$
I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1},
$$
where $ F(z, s) = \dfrac{\sinh(\sin^2[π\Gamma(z)/(2z)])...
- 375
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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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27
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How does the principal value affects to the limit here?
In Córdoba and Gancedo - Contour dynamics of incompressible 3-D fluids in a porous medium with different densities (page 4) I read that if
$$ v (x_1,x_2,x_3,t)=-\frac{\rho_2-\rho_1}{4\pi} \...
- 383
1
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0
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303
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Continuity of the Legendre transform of a Lipschitz function
Let $p > 1$, $f$ defined on $L^p(\mathbb{R}^n,\mathbb{R})$, locally Lipschitz and concave, with $f(x) = 0$ when $x \geq 0$ a.e. We define, with $q$ dual to $p$, for any $y\in L^q$ , $g(y) := \sup_{...
- 31
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91
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Gaussian width and restricted isometry
It is known that, for an $m$ dimensional space and an $n\times m$ dimensional random matrix $U$ whose entries are iid Gaussian, then $\|I-(1/n)U^TU\|$ is bounded by $\sqrt{m/n}$ when $n>m$.
If a ...
- 11
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126
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Almost every where divergent Fourier series
Does there exist any continuous function $f:[\pi,\pi]\to \mathbb{C}$ whose Fourier series $\sum \hat{f}(n)e^{int}$ is almost every where divergent?
- 4,058
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76
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Existence of a `right' sequence
Suppose that $f_n: \mathbb{R} \times [0,\infty) \to \mathbb{R}$ is a sequence of bounded variation functions such that there exists $C>0$: $|f_n| < C$ for every $n$ and, as $n \to \infty$, for $...
- 41
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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
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87
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An oscillatory integral estimate
Let $n \geq 3$ and consider two sequences of strictly monotone functions $\{\mu_l(t)\}_{l=1}^{n}$ and $\{\lambda_l(t)\}_{l=1}^n$ on the interval $[-1,1]$ with $\mu_l(0)=0$ and $\lambda_l(0)=1$ for all ...
- 4,143
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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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86
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Coboundary in the slow mixing systems
Given dynamical system $(X, T, \mu)$, $\mu$ is probability, $\mu \circ T =\mu$, $T$'s transfer operator $P$ is defined by following relation: $\int (P a) \cdot b d\mu= \int a \cdot (b \circ T) d\mu$ ...
- 553
1
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0
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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
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78
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Potential for a Monotone Operator
[Cross-posted from math.stackexchange]
I have a question about understanding the proof of Theorem 4.11 in the paper A Potential Theory for Monotone Multivalued Operators (accessible here). The ...
- 291
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389
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The perturbation of a convex function can also be convex?
$ W^{1,\infty}(D)\ni f:D\to\mathbb R, (x,y)\mapsto f(x,y)$, is a strictly increasing on both dimensions (i.e. if $x_1>x_2$ then $f(x_1,y)>f(x_2,y)$), lipschitz continuous function defined on a ...
- 263
1
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0
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49
views
On different norms of the interpolating operator
Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
- 1,209
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922
views
A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it
Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
- 698
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74
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Nonlinear maps in Riesz Thorin theorem
The Riesz Thorin theorem allows us to interpolate between $L^p$ spaces and the usual assumption is that the map $T$ is linear.
What I was wondering about is whether this is because otherwise you do ...
user127450
1
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0
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93
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Relative boundedness of the adjoint
Let $X$ be a separable Banach space and $T_1:D(T_1) \subset X \rightarrow X$ and $T_2:D(T_2) \subset X \rightarrow X$ two closed operators with $D(T_2)\subset D(T_1)$ and $D(T_2^*) \subset D(T_1^*).$
...
- 33
1
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0
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211
views
Propagation of singularities and the Schrodinger equation
I always thought that the propagation of singularities theorem by Hörmander says (on $\mathbb R^n$ for a classical symbol $p(x,\xi)=\xi^2+V(x)$) that for a Schrödinger equation
$$(i \partial_t-p(x,D))...
- 11
1
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0
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45
views
Shifting Sobolev norms in a hyperbolic estimate
Suppose $\Omega$ is a bounded domain and $\omega \subset \Omega$. Suppose we have the following estimate:
$$ \|u\|_{H^1((0,T) \times\Omega)} \leq C (\|u\|_{H^1((0,T) \times \omega)} + \|\Box u\|_{L^2((...
- 4,143
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126
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identity involving spectral functions
Let $A$ be any compact operator and let $A^*$ denote its adjoint. Let $f$ be a spectral function. Then is the following true :
$$ A^* f(AA^*) = f(A^* A) A^*$$
- 519
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237
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On the bound of the Stein-Wainger oscillatory integral
Let $\lambda\in \mathbb{R}$, $\phi\in C^\infty(\mathbb{R})$. We define the Stein-Wainger oscillatory integral by
$$I=p.v.\int_\mathbb{R} e^{i\lambda\phi(t)}\frac{dt}{t}.$$
Stein-Wainger [1] showed ...
- 11
1
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0
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50
views
Comparison of (square) of a function and its Fourier transform in an integral
I am completely stuck on a comparison between $f(t)^2$ and $\hat{f}(t)^2$ in an integral.
Considering $f(t)$ of rapid decrease at infinity such that near zero: $f(t) \sim_0 t^{-\frac{1}{2}- \alpha}+o(...
- 1,199
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0
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170
views
non-analytic functions with arbitrary large derivatives [closed]
This may be a trivial question but I can't see it immediately.
Suppose $\{a_k\}$ is an increasing sequence of positive reals. Does there exist a smooth function $f \in C^{\infty}([0,1])$ such that $\...
- 4,143
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0
answers
101
views
Non standard Lipschitz extension
Consider a ball B and let $f(x) \in L^1(B)$ such that $\int_B f(x) dx = 0$. Furtheremore, there exists a closed set $E \subset B$ such that $f|_E$ is Lipschitz. The standard Lipschitz extension ...
- 455
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0
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137
views
Is there an analysis theorem analogous to Kuznetsov/Petersson trace formula?
I am thinking about general differential operator acts on a compact manifold. Is there something similar to Kuznetsov trace formula?
For example, let $f_i $ be the eigenfunctions of an operator $D$, ...
- 3,804
1
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1
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228
views
Which norms on vectors can be consistently decomposed?
I need to know which permutation-invariant norms can be consistently decomposed in the sense that for any vector $v = (a,b,c)$ we have that
$$\|(a,b,c)\| = \|(\|(a,b)\|,c)\|.$$
More precisely, let $v ...
- 702
1
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1
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139
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Compactly supported functions and projections
Let $\Omega$ be an open subset of $\mathbb{R}^n$ and take a family of continuous compactly supported functions $f_n$ on $\Omega$ normalized to one (in the $L^2$ sense).
Then, these functions span a ...
- 501
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0
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124
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Inequality about the Fourier transform: $\Vert u \Vert_{L^k} \le \Vert \mathcal{F}(u) \Vert_{L^m}$ (where $1 \le m \le 2$ and $m,k$ Holder conjugates)
How can I prove the following inequality about the Fourier transform?
$$\Vert u \Vert_{L^k(\mathbb{R}^N)} \le \Vert \mathcal{F}(u) \Vert_{L^m(\mathbb{R}^N)}$$ for $1 \le m \le 2$ and $m,k$ Holder ...
user89890
1
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0
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76
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Which sets support which spectra?
I know (and this is of course rather elementary) that an isolated point in the spectrum of a self-adjoint operator $T$ always belongs to the point-spectrum.
I would like to ask: Are there similar ...
- 173
1
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0
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186
views
Using continuity + commutativity to define "limit"
Let $S$ be the set of real sequences, and $S_0\subset S$ be the set of convergent real sequences. For each $f:\mathbf R\to\mathbf R$, we define $\bar{f}:S\to S$ by $$\bar{f}(\{x_n\})=\{f(x_n)\}.$$
It ...
- 1,630
1
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0
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105
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Generalize characterization of upper semicontinous functions
Let $X$ be a metric space and denote $f:X \rightarrow \mathbb{R}.$
It is easy to show that the following two statements are equivalent:
$(1)$ For any real number $c$, we have $f^{-1}(-\infty,c)$ is ...
- 623
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1
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162
views
Does there exist a class of real-valued upper semicontinuos functions on $X$ such that $\mathcal{F}$ is countable?
Ian Morris quoted the following:
For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set $\mathcal{F}\...
- 623
1
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0
answers
180
views
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 ...
- 115
1
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0
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105
views
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
1
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0
answers
194
views
Cotlar-Stein's Lemma and the Dirichlet kernel
It is well-known that Cotlar-Stein's Lemma can be used to prove the $L^2$ boundedness of the Hilbert transform. See e.g. $L^2$ boundedness of the Hilbert transform via Cotlar-Stein Lemma. Then using ...
- 171
1
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0
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331
views
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
1
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0
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128
views
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 $...
- 31
1
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0
answers
123
views
Generalization of concave envelope
Let $g:\mathbb R_+\to\mathbb R$ be a measurable function (which could be supposed to be bounded and Lipschitz if required). Let $\mathcal P$ be the collection of probability measures $\mu$ on $\mathbb ...
- 1,835
1
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1
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238
views
Does the bounded extension of the Fourier multiplier operator agrees with its original explicit definition?
We consider the Fourier multiplier operator $T_0$ defined by the explicit expression
$$(T_0f)(x)=\int_{\mathbb{R}^n}{e^{ix\cdot \xi}m(\xi)\hat{f}(\xi)d\xi}, \ f\in S(\mathbb{R}^n),$$ where $S(\mathbb{...
- 171
1
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0
answers
86
views
Least restrictive condition such that $-y''(x)+q(x)y(x)=\lambda y(x)$ has two solutions
Let $\lambda \in \mathbb{R}$ and consider some function $q$ on the interval $[0,1].$
What is known to be the least restrictive condition on $q$ such that there are two linearly independent $H^2$ ...
- 11
1
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0
answers
117
views
The eigenfunction of modified $1$-laplace equation?
Let $\Omega\subset \mathbb R^2$ be open bounded with smooth boundary. It is well known that the laplace equation
$$
-\Delta u=0
$$
has a set of eigenvalues $0<\lambda_1<\lambda_2\leq\lambda_3<...
- 679
1
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0
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84
views
extension for a complex operator
Let be $\lambda>0$. Put
$$ L_{\lambda}=\Big[-\frac{\partial^{2}}{\partial z \partial \overline{z}}+\lambda^{2}|z|^{2} +\lambda\Big(\overline{z}\frac{\partial}{ \partial \overline{z}}-z\frac{\...
- 135