# Tagged Questions

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### Typo in Caffarelli-Silvestre?

I am reading two papers by Caffarelli and Silvestre, namely Regularity results for fully nonlinear equations by approximation and The Evans-Krylov theorem for nonlocal fully nonlinear equations. From ...
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### Di Perna-Lions theory for transport equation

Does someone know if some notes on the topic mentioned in the title are available online? I'm reading the paper "Ordinary differential equations, transport theory and Sobolev spaces" by Di Perna and ...
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### Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$?

Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$, where $H^1(R;H^1(R^n))=\{f\colon R\to H^1(R^n)\colon \int \|f(t,\cdot)\|_{H^1}^2\,dt <\infty \quad \text{and} \int \|f'(t,\cdot)\|_{H^1}^2\,dt <\infty \}$ ...
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### Weakly * continuous definition

What does it mean that $$t\to u(t,\cdot)$$ is waakly* continuous from $[0,T]$ to $L^\infty(\mathbb{R}^d)$? I guess that I have to see $L^\infty(\mathbb{R}^d)$ as the dual of $L^1(\mathbb{R}^d)$ and ...
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### Green's representation on a compact domain

This is from page 17-18 of Trudinger and Gilbarg Let $\Omega$ be a domain for which the divergence theorem holds. Let $\Gamma(x-y)$ be the normalised fundamental solution of the Laplace's equation, ...
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### Deriving lower bound for eigenvalues of laplace operator

Let $u$ be a function such that $$\Delta u + \lambda u = 0 \quad \mbox{ on } \quad D, \quad u = 0 \quad \mbox{ on } \partial D$$ and let $w$ be a function such that $\Delta w + \beta w < 0$ ...
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### Show that $L[v^2] := \Delta(v^2) + \frac{2}{w} \sum_{i=1}^n w_{x_i} (v^2)_{x_i} \ge 0$

Let $u$ be a function such that $$\Delta u + \lambda u = 0$$ for some $\lambda \in \mathbb R$, also let $w$ be a function such that $$\Delta w + \beta w < 0.$$ for some $\beta \in \mathbb R$. ...
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### Constructing a specific normalised test function.

Given the following class of normalised test functions: \mathcal{T}=\{\phi\in C_c^{\infty}(\mathbb{R}^n)\ |\ \text{supp }\phi \subseteq B(0, 1),\ \|D\phi\|_{\infty}\leq 1\} ...
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### A priori estimates for functions in $C_0^\infty (\overline{\Omega})$.

Let $u\in C_0^\infty (\overline{\Omega})$, where $\Omega\subset \mathbb{R}^N$ is a bounded domain. Fix some $a\in \Omega$ and choose $r>0$ such that $\overline{\Omega}\subset B(a,r)$. Define ...
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### Explicit solution of the nonlinear Schrödinger equation

Consider the linear Schrödinger equation, $$(LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases}$$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
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### The spherical mean of $h(x,y) = x$.

The spherical mean of a function $h : \mathbb R^2 \to \mathbb R$ is given by $$\frac{1}{2\pi r} \int_0^{2\pi} h(x + r \cos(\theta), y + r \sin(\theta)) d \theta$$ Now I want to compute the ...
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### Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponenent allows for convenient ...
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### Solve PDE by getting two ODEs

My goal is to solve this PDE for $f:[-1,1] \times \mathbb{R}_{\ge 0}\rightarrow \mathbb{C}$ $$\partial_t f(x,t) = -\partial_x^2 f(x,t) + g(t)V(x)f(x,t).$$ I would consider this PDE to be solved if ...
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### Inverse Laplace operator $\Delta^{-1}$ and Sobolev spaces

I'm looking for some regularity results for the inverse Laplace operator. More precisely - we're set in $\mathbb{R}^3$ and we are looking at the operator $$\Delta^{-1}f = \frac{x}{|x|^3} \ast f$$ I'd ...
I want to solve the 2 dimensional heat equation on a square $\Omega = \{ (x,y) : 0 < x < \pi, 0 < y < 2\pi \}$ with the Fourier Method \begin{align*} \partial_t u - \Delta u & = 0 ...