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Results tagged with ap.analysis-of-pdes
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user 68232
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
1
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0
answers
60
views
$L^p$ estimates for critical heat equation on $\mathbb{R}^n$
Background:
Let $u:\mathbb{R}^n\times [0,T_+)\to \mathbb{R}$ be a solution to
$$\partial_t u = \Delta u + |u|^{p-1}u$$
where $p=2^{*}-1=\frac{2n}{n-2}-1=\frac{n+2}{n-2}$, $n\geq 3$ and $T_+>0$ denotes …
- 557
4
votes
1
answer
135
views
Hölder Gradient Estimates for Linear Elliptic Equations in higher dimensions
I was looking at Theorem 12.4 of Gilbarg and Trudinger's Elliptic partial differential equations of second order (MR1814364, Zbl 1042.35002):
Theorem 12.4. Let $u$ be a bounded $C^2(\Omega)$ solution …
- 557
0
votes
1
answer
61
views
Explicit expression for $(-\Delta)^s (|x|^2)$ where $x\in \mathbb{R}^n$ ($n>2s$) and $s\in (...
Do we know an explicit expression for $(-\Delta)^s (|x|^2)$ where $x\in \mathbb{R}^n$ ($n>2s$) and $s\in (0,1)?$
- 557
1
vote
0
answers
172
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Converting equation on the sphere to $\mathbb{R}^n$
Consider the following equation $-\Delta_{\mathbb{S}^n} u = u$ where the $(\mathbb{S}^n,g)$ where $g$ is the usual metric induced by the inverse stereographic projection $S:\mathbb{R}^n\to \mathbb{S}^ …
- 557
2
votes
0
answers
68
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Fundamental solutions for weighted laplace equation
Consider the equation $L_w u = \frac{1}{w}\operatorname{div}(w\nabla u) =f(x)$, on $\mathbb{R}^n$ with radial weights $w(x)=w(|x|).$ Then I am interested in the fundamental solutions for the operator …
- 557
0
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1
answer
132
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An inequality involving weight $|x|^\alpha$
Background:
Let $u:\mathbb{R}^n\to \mathbb{R}$. Then the paper considers the following problem
\begin{align*}
-\operatorname{div}(w(x) \nabla u) &= w(x) \text{ in } \Omega \\
u &= …
- 557
0
votes
1
answer
75
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Reverse Hölder type inequality for the Laplacian raised to a power
I am studying integrals of the form $\int (\Delta \rho)^{\alpha} f^{\beta}$ where $0<\alpha < 1, \beta \geq 0$ and $\rho, f \in C_c^{\infty}(\mathbb{R}^n).$ My goal is to obtain lower bound on this qu …
- 557
2
votes
0
answers
183
views
Are solutions to this elliptic PDE uniformly bounded in $\mathbb{R}^n?$
Given a fixed value $\lambda>0$ let $u\in \dot{H}^1(\mathbb{R}^n)$ be a weak solution (or eigenfunction with eigenvalue $\lambda$) in dimension $n\geq 3$ to the following PDE,
$$-\Delta u = \lambda \r …
- 557
2
votes
0
answers
136
views
Computing the fractional laplacian of a logarithm function
Are there explicit formulas to compute the fractional laplacian $(-\Delta)^{s/2}\log |x|$ in $\mathbb{R}^n$ for $s\in (0,2)$?
- 557
2
votes
1
answer
99
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How to estimate higher order regularity for wave type equation with time dependant coefficie...
Consider the following wave-type equation,
$$u_{tt}-\frac{2}{t}u_t-\Delta u=g(t,x)$$
where $(t,x)\in [\epsilon, 1]\times \mathbb{R}^3$ for some $\epsilon>0.$ Furthermore assume that $(u,u_{t})=(0,0)$ …
- 557
0
votes
0
answers
396
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Sobolev inequality on the sphere derivation
I am reading the following paper (preprint here) and the author starts by stating the Sobolev inequality on the Sphere $\mathbb{S}^d$
$$\frac{p-2}{d}\int |\nabla u|^2 + \int |u|^2 \geq \left(\int |u| …
- 557
1
vote
0
answers
127
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Laplacian on the sphere and Moving Plane method
Consider the sphere $\mathbb{S}^n$ as a subset of $\mathbb{R}^{n+1}$, thus $\mathbb{S}^n=\{\omega\in \mathbb{R}^{n+1},\sum_{i=1}^{n+1}\omega_i^2=1\}.$
I am interested in studying positive solutions to …
- 557
0
votes
0
answers
59
views
How to prove this estimate involving the Stein Derivative?
Recall the Stein Derivative,
$$\mathcal{D}^{s} f(x)=\left(\int \frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2 s}} d y\right)^{1 / 2}.$$
I want to show that,
$$\left\|\mathcal{D}^{s}(f g)\right\|_{2} \leq\left\|f \ …
- 557
1
vote
1
answer
141
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Extremizers of the Sobolev inequality
Background:
I am reading the paper: Best constant in Sobolev inequality by Talenti (see here) and I am trying to understand the following step.
On p. 365, the author is arguing that the solutions to t …
- 557
3
votes
1
answer
560
views
Does the following version of the Coifman–Meyer Theorem exist?
Recall the Coifman–Meyer Theorem as stated in Grafakos and Oh - The Kato-Ponce Inequality.
Theorem: Let $m \in L^{\infty}\left(\mathbf{R}^{2 n}\right)$ be smooth away from the origin and satisfy
$$
\l …
- 557