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      All Questions

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      62 views

      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 $...
      1
      vote
      0answers
      93 views

      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}\...
      2
      votes
      1answer
      66 views

      Ground state for non-linear Schrödinger

      When studying the blow-up for focusing non-linear Schr?dinger equation (NLS) one often compares the initial-state to a stationary solution. In the energy-critical case, this stationary solution is ...
      3
      votes
      2answers
      171 views

      Graph with complex eigenvalues

      The question I am wondering about is: Can the discrete Laplacian have complex eigenvalues on a graph? Clearly, there are two cases where it is obvious that this is impossible. 1.) The graph is ...
      -1
      votes
      1answer
      60 views

      Is this kind of interpolation correct?

      Let $f=\sum f_j$ be a finite sum. Assume that $$ \|f\|_2\le(\sum\|f_j\|_2^2)^\frac12$$ $$\|f\|_\infty\le C\max_j\|f_j\|_\infty$$ Then can we conclude that for $2<p<\infty$ $$\|f\|_p\le C^{1-\...
      1
      vote
      1answer
      129 views

      Continuity of subharmonic functions

      There is a result saying that the set where a subharmonic function defined on an open set of $\mathbb{R}^{m}$ ($m\geq2$) is discontinuous is a polar set. Could someone give me a reference for this ...
      1
      vote
      0answers
      130 views

      Lax Milgram for non coercive problem?

      I obtained the variational form of my problem. and the bilinear form is below. Bilinear Form Let $\Omega\subset\mathbb{R}$ be an open set. For $u,v\in H^1_0(\Omega)$. I have $$a(u,v)=\int_\Omega u(t)...
      2
      votes
      2answers
      140 views

      Rate of convergence of mollifiers // Sobolev norms

      Following up to the question raised here, I am searching for a reference (or a simple argument) to establish (in the whole space) the following (suggested) equivalence : Given $N_1$ and $N_2$ two (...
      0
      votes
      3answers
      126 views

      Does the Leibniz (product) rule hold for the spectral fractional Laplacian?

      Does the Leibniz (product) rule hold in some sense for the spectral fractional Laplacian (at least in 1 dimension)?
      0
      votes
      0answers
      47 views

      Does an integration by parts formula hold for the spectral fractional Laplacian in 1-d?

      Is there an integration by parts formula for the spectral fractional Laplacian in a bounded interval $[a,b] \subset \mathbb R$?
      0
      votes
      0answers
      26 views

      Smooth compactly supported function with good scaling with respect to the fractional Laplacian

      Is there a smooth cut off function with compact support such that $\phi: \Omega \subset \mathbb R^N \to \mathbb R$, $\mathrm{supp} \phi \subset B_R(0) \subset \Omega$ and $$(-\Delta)^s \phi \le C R^{-...
      1
      vote
      1answer
      153 views

      Space derivative of flow of ODE with monotone source

      Consider the ODE $$ \begin{cases} \partial_t\Phi(t,x) = f(t,\Phi(t,x)), &\ t>0, \ x \in \mathbb R \\ \Phi(0,x) = x, & x \in \mathbb R \end{cases} $$ where $f$ is function which is a non-...
      5
      votes
      1answer
      135 views

      Ratio of integrals with increasing dimension over Euclidean balls

      Let $f_n(x)\geq0$ be any sequence of nonnegative $L^1(\mathbb{R}^n)$ functions such that $\int_{\mathbb{R}^{n}}f_n(x)dx=1$ where $dx$ is the Lebesgue measure on $\mathbb{R}^n$. For any $a>1,\...
      2
      votes
      2answers
      160 views

      A Fredholm equation with a particular kernel

      How to solve this kind of Fredholm’s equation? $$ x(t)+\lambda \int\limits_{0}^{1}\! \big[ts - \min\{t,s\}\big]x(s)ds=t $$ Thanks for any help.
      6
      votes
      1answer
      263 views

      Sub-Gaussian decay of convolution of $L^1$ function with Gaussian kernel

      I think it might be helpful to put the new statement at the beginning and put the original post at the end. This new statement is more mathematically elegant. Let $f\geq0$ be in $L^1(\mathbb{R}^d)$ ...

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