Questions tagged [ap.analysis-of-pdes]
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
10 questions from the last 7 days
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Linear and non-linear intersection to solve ODE
Consider a linear operator $$L(u(t)) = \dfrac{d}{dt}u(t)+p(t)u(t)$$ for known function $p(t)$. It is well known the homogeneous equation $$L(u) = 0 ~~\text{or}~~\dfrac{d}{dt}u(t)+p(t)u(t)= 0$$ has ...
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32
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Examples of subharmonic functions
Let $A$ be a constant symmetric matrix with $\lambda < A < \Lambda$ and $0<\lambda < \Lambda$ are fixed constants. Let $u$ be a solution of $\text{div}A \nabla u = 0$. Is it true that $\...
- 455
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30
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Explicit rate of decay of the positive standing wave of the subcritical nonlinear Schrödinger equation
Consider the following semilinear problem:
$$
\begin{cases}
- \Delta u + u = u |u|^{p - 2}
&\text{in} ~ \mathbb{R}^N;
\\
u (x) \to 0 &\text{as} ~ |x| \to \infty,
\end{cases}
$$
where $N \geq 2$...
- 144
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16
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Third order estimate for linear elliptic equations
Let $\lambda < A < \Lambda$ be a constant symmetric matrix and $u$ be a $C^{\infty}$(elliptic regularity gives smooth solutions) solution of $\text{div} A \nabla u = 0$. Let $S_1$ be a sphere ...
- 455
2
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1
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70
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Tangential Sobolev spaces
Let $Ω⊂R^n$ be a smooth domain, define $U_s=\{x∈Ω | d(x,∂Ω)<s\}$; let $f∈W^{1,p}(Ω)∩W_{\mathrm{loc}} ^{2,p}(Ω)$; let $v$ be the unit normal to $Ω$; consider $v$ to be smooth with bounded ...
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13
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On compact embeddings in weighted Riesz potential spaces
I wonder if there is any references for the study of the following type of spaces
$$ X_{\delta,\alpha}=\{ u\in L^2_\delta(\mathbb{R}^n):\, u= (-\Delta)^\alpha f \quad\text{for some}\quad f\in L^2_{\...
- 4,135
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47
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Sobolev embedding related to the Grushin operator
Consider $\mathbb{R}^{N} = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$ and a vector in $\mathbb{R}^N$ will be denoted by $z = (x,y)$. Also take $\Omega_i \subset \mathbb{R}^{N_i}$ a bounded domain and $\...
- 187
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85
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+100
Uniqueness of bubbling points in Struwe's global compactness theorem
I am reading the following paper of Struwe in which he proves the following result:
Proposition 2.1:
Let $n\geq 3$, $\lambda \in \mathbb{R}$ and $\Omega$ be a smoothly bounded domain in $\mathbb{R}^{n}...
- 537
4
votes
1
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Mapping properties of the Schrödinger semigroup
The Schrödinger semigroup $e^{t(-\Delta +V(x))}$ for Kato class potentials is fairly well-understood. A classical reference is the AMS paper "Schrödinger Semigroups" by Barry Simon. I was ...
- 1,045
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221
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A deceptively simple regularity problem for functions on the plane
By various meanderings and toying with simpler problems, my current research has lead me to the following quite straightforward question, which I am wholly unable to answer:
Consider a twice ...
- 989