All Questions
Tagged with real-analysis ap.analysis-of-pdes
569 questions
1
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Bound for a certain integral expression
I am working to establish an estimate in $X^{s,b}$ spaces to prove local well-posedness of a certain equation, and I need to consider some sub-cases. In particular, I wish to show that the following ...
- 41
4
votes
2
answers
3k
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Chain rule for fractional laplacian
Does anyone know a formula of chain rule for fractional laplacian?
say we take the fractional laplacian of order a on function $g(U(x))$ $x\in \mathbb{R}^2$, $U \in \mathbb{R}$, $g \colon \mathbb{R} \...
- 459
1
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1
answer
163
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Pohozaev result for equations with weights
I am interested in nonnegative solutions of
$-div( e^{-\gamma(x)} \nabla u(x)) = e^{-\gamma(x)} u(x)^p$ in $\Omega$ with $ u=0$ on $ \partial \Omega$.
Or instead the equation $ -\Delta u + \...
- 2,494
3
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1
answer
495
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Inequality in the Sobolev space $H^1$
I've found the following inequality
$$\int_{B_r}\vert u\vert^q\leq C \bigg(\int_{B_r}\vert\nabla u\vert^2\bigg)^{a}\bigg(\int_{B_r}\vert u\vert ^2\bigg)^{\frac{q}{2}-a}+\frac{c}{r^{2a}}\bigg(\int_{B_r}...
- 2,933
14
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633
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Classes of (non-continuous) functions with the fixed point property
Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...
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- 1
2
votes
2
answers
283
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A general inequality about spherical mean of a function
suppose $\overline u(r)=\frac{1}{\omega_{n-1}}\int_{S^{n-1}}u(r,w)dw,0<r<1,$ is the average of $u(r,w)$ on sphere $S^{n-1}$,where $(r,w)$ are the polar coordinates in $R^n$.
My question is ...
- 2,494
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0
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267
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Extension of solutions of PDE
Let $\Omega \subset \mathbb{R}^{2}$ be an open set such that $\mathbf{0} \in \Omega$. Let $A := \Omega \setminus (\{0\}\times \mathbb{R})$, that is, $A$ is $\Omega$ with the $y$-axis removed.
Let $F:...
- 11
2
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1
answer
208
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Does a particular iteration produce a weak solution to a non linear pde?
Consider the following non linear pde in the unknown $v(x,y)$:
$$ \frac{\partial v(x,y)}{\partial x} +
\Big(\frac{\partial v(x,y)}{\partial x} \Big)^2 = e^{2 ty}-1 $$
where $t$ is some fixed small ...
- 16.2k
3
votes
2
answers
2k
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The extension of smooth function
If $U$ is a bounded domain in $\mathbb R^n$ whose boundary is smooth, and $f$ is a smooth function on $U$ whose partial derivatives of all orders have a continuous extensions to $\bar U$. For an ...
- 31
1
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2
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938
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Alternate definitions of $C^{1,\alpha}$ and $C^{1,\alpha}(\bar{D})$ maps
My question is about the precise definition regarding the following:
Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a ...
- 60.5k
5
votes
0
answers
428
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Is there an appropriate weighted Sobolev space to include exponential map and projection map?
Observe that given a non negative function
$\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted
$L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions
$f: \...
- 3,245
4
votes
1
answer
261
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Minimizing action squared versus action
I have a very basic question in the calculus of variations:
Suppose I want to minimize the functional
$$A[r, r'] = \int_\Omega L(r, r') dx $$
When is it possible to say that extremals of $A$ agree ...
- 614
4
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0
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340
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Viscosity solution of the PDE
Let $\Omega$ be bounded domain, $u=0$ on $\delta\Omega$ and
$$|Du|-f(x,u)=0$$
where $f\ge 0$ and $f$ is strictly monotone for fixed $x.$ I am looking for the reference to show that it has unique ...
- 61
3
votes
1
answer
258
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Subharmonic envelope
I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me know....
- 2,270
1
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1
answer
199
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On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis
Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. Then my question is :
...
- 91.8k
3
votes
2
answers
175
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Decay rate of nonlocal differential operator?
Hi, Moers.
Let $m(\xi) \in S^0$, that is,
$$
|D^\alpha m(\xi)| \leq C<\xi>^{-|\alpha|}, \quad \forall \xi \in R^n.
$$
It's well known that $m(D)$ is bounded in $L^p$ for $1 < p < \infty$.
...
- 425
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1
answer
612
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Calculating a distributional derivative
Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to ...
- 34.7k
2
votes
2
answers
408
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Higher order partial derivatives and global regularity.
Let $f$ be a function of two variables $x$ and $y$. Assume that $f$ is $C^1$. Assume that $f_{xx}$ exists and continuous.
Is it true that $f_{xy}$ exists and continuous?
Is it true that $f_{yx}$ ...
- 314
9
votes
1
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782
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Mean value property with fixed radius
Let $f$ be a continuous function defined on $\mathbb{R^n}$. It is well known that both the spherical mean value property (MVP) of $f$, i.e.
$$f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}f,\ ...
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