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Results tagged with ap.analysis-of-pdes
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user 112207
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
vote
0
answers
62
views
Does sets of positive capacity rule out constant functions?
Let $U\subset \Bbb R^d$ be bounded with Lipschitz boundary $K\subset \bar{U}$ be compact. The capacity of $K$ in $U$ is defined by
\begin{align*}
\text{Cap}_{p}(K, U) :=
\inf \left\{
\int_U |\n …
- 1,101
2
votes
0
answers
144
views
Positivity of the Fourier transform: prove or disprove that $\operatorname{Re}(\overline{\wi...
Let $F:[0,\infty) \to[0,\infty)$ be increasing, $C^1$ and $L-$Lipschitz with $F(0)=0$. Let $u\in L^1 (\Bbb R^d)$, $u\geq0$ so that $F\circ u\in L^1 (\Bbb R^d)$
I would like to prove (or disprove) tha …
- 1,101
1
vote
0
answers
54
views
Comparison principle for porous medium equation in Fourier variables
Let $V:[0,\infty) \to[0,\infty)$ be convex, $C^2$ with $V(0)=0$. Define $F(u): = uV'(u)-V(u) $. Let $v\in L^1 (\Bbb R^d)$, $v\geq0$ so that $F\circ v\in L^1 (\Bbb R^d)$. For fixed $\varepsilon>0$, as …
- 1,101
7
votes
1
answer
341
views
Does the pointwise mean value property imply harmonicity?
Assume $u:\Omega\subset\mathbb{R}^d\to\mathbb{R}$ is continuous and satisfies the property:
for every $x\in \mathbb{\Omega}$ there is $r_x>0$ such that
$$
u(x)=\frac{1}{|B(x,r_x)|}\int_{B(x,r_x)} u(y) …
- 1,101
5
votes
1
answer
447
views
Seeking for references on some PDEs
This is not a technical mathematical question. I came across some PDEs with no references nor their names.
$$-\Delta u + \int_\Omega udx = f\qquad \hbox{in $\Omega$} \label{1}\tag{Eq1}$$
The above equ …
- 1,101
4
votes
2
answers
223
views
Looking for a reference or the procedure on how to solve the parabolic equation with $L^2$-w...
Let $\zeta, u_0\in L^2(\Omega)$, with $\zeta \geq 0$ and $\Omega\subset \Bbb R^d$ open and bounded.
\begin{equation}\label{Star-3.7}
\begin{cases}
\partial_t u -\Delta u + \zeta u=0 &\mbox{ in }\; \ …
- 1,101
5
votes
0
answers
135
views
Measure of the boundary of an BV-extension domain: do we have $|\nabla Eu|(\partial \Omega)=0?$
Let $\Omega\subset \Bbb R^d$ be open. The space $BV(\Omega)$ consists in functions $u\in L^1(\Omega)$ with bounded variation, i.e. $|u|_{BV(\Omega) }<\infty$ where
\begin{align}\label{eq:bounded-var …
- 1,101
3
votes
0
answers
111
views
Look for a suitable cut-function: from Pierre Grisvard "Elliptic Problems in Nonsmooth Domai...
From Pierre Grisvard "Elliptic Problems in Nonsmooth Domains":
Theorem[Theorem 1.4.2.1]
Let $\Omega$ be an open subset of $\mathbb{R}^d$ with a continuous boundary, then $C_c^\infty(\overline{\Omega …
- 1,101
3
votes
2
answers
393
views
Is a bounded sequence of $H^1(\Omega)$ tight?
Assume $\Omega$ is a bounded subset of $\Bbb R^d$ and $ (u_n)_n$ is a bounded sequence of the Sobolev space $H^1(\Omega)$.
Question: Can we say that $ (u_n)_n$ is tight in $L^2(\Omega)$ namely: …
- 1,101
1
vote
0
answers
62
views
Regularity of a shrunken domain
I am encountering a geometrical question that intuitively seems obvious but I have a lack of argument to prove or disprove it in a rigorous manner.
Let $\Omega\subset\Bbb R^d$ be an open bounded (ma …
- 1,101
10
votes
1
answer
1k
views
Global regularity for Neumann problem
Let $\Omega\subset \mathbb{R}^d $ be a bounded open subset ($d\in \mathbb{N}$) and denote $\partial\Omega$ its boundary which we assume to be Lipschitz. The classical inhomogeneous Neumann problem …
- 1,101