New answers tagged ap.analysis-of-pdes
2
votes
Accepted
Well-posedness of the linear parabolic equations with respect to the inhomogeneous term as well as the initial data
Edited to make it correct for future reference.
You might want to look into this expository paper on maximal regularity for linear parabolic equations:
https://people.math.ethz.ch/~salamon/PREPRINTS/...
2
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Let $u$ be harmonic on domain $D\subset \mathbb R^d$, how far can we extend $u$ holomorphically?
See a similar question I asked some times ago. Due to P. Lelong and J. Siciak it is known that, if $D\subset\mathbb R^n$ ($n\geq 2$) is a domain, then there exists a domain $\tilde D\subset\mathbb C^...
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2
votes
Nature of a certain invariant on smooth field of positive definite matrices
I don't see where $g$ being SPD is used? The content of this question boils down to: given $F:\mathbb{R}^n\to\mathbb{R}^n$, I want to know whether there is a $n\times n$-matrix valued function $A:\...
- 35.4k
1
vote
Accepted
Exterior differential systems on compact three-manifolds and Cartan-Kähler theory
In their comments to my first answer, the OP has clarified that they did not mean to regard the metric $h$ as a given, but, rather, an output of the problem of prescribing coframings by specifying ...
- 106k
3
votes
Exterior differential systems on compact three-manifolds and Cartan-Kähler theory
Let me phrase the problem as I understand the given data and then describe how the 'theory of exterior differential systems' would be applied.
One starts with a compact Riemannian $3$-manifold $(M,h)$,...
- 106k
5
votes
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Stochastic representation of Laplace equation with Neumann boundary condition
Yes, see Section 4.4.2 in "Stochastic Differential Equations, Backward SDEs, Partial Differential Equations" by Pardoux and Rascanu.
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4
votes
Accepted
Can we approximate a Hölder pdf by higher-order Hölder pdf's?
No, take any $p$ such that $p(x)=|x|^\alpha$ in some neighbourhood of the origin.
- 8,464
8
votes
Wick rotation for Laplace and wave equations
The transformation to imaginary time is used to relate the Green's function of the Laplacian to the Green's function of the d'Alembertian (therefore relating Laplace equation and heat equation). See ...
- 172k
6
votes
Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces
There's a famous trick attributed to Minty (sometimes also to Browder and Lions) which allows to recover at least a.e. convergence when you're able to pass to the limit through an increasing non-...
- 2,505
4
votes
Accepted
Reference request: Uniformly elliptic partial differential operator generates positivity preserving semigroup
Check Chapter 4, more specifically Chapter 4.2 in Ouhabaz: Analysis of heat equations on domains (2005). ZBL1082.35003. The approach there goes by the form method which I personally perceive to be ...
- 2,100
0
votes
If $t \to \lVert f(\cdot,t) \rVert_{L^2_x}^2$ is absolutely continuous, can we interchange the spatial integral and time derivative? (from MSE)
I will not assume function $$\psi(t) := \lVert f(\cdot,t) \rVert_{L^2_x}^2$$ be absolutely continuous.
We have $f\in W^{1,q'}_t\bigl([0,\infty), L^{p'}_x(S^1)\bigr)$, so $f$ has a representative $\...
- 399
0
votes
Reference: Result of interior parabolic regularity theory for Hamilton–Jacobi equations
I only found the result for $p=2$
Herbert Amann, Existence and stability of solutions for semi-linear parabolic systems, and applications to some diffusion reaction equations, Theorem 3.1.
Since the ...
0
votes
Comparison of solutions of Hamilton-Jacobi equations with different initial conditions
Consider the general case
$u_t+H(x,t,u_x)=0$ in $\mathbb R^n\times [0,T],$
where $H(x,t,p)$ is uniformly continuous for bounded $p$. The desired result holds when we have
$|H(x,t,p)-H(y,t,p)|\leq m(|x-...
2
votes
Accepted
Decompositions of $\partial_i$ to the radial direction and rotations in higher dimensions
The formula you gave in the beginning is a special case of the vector triple product formula in $\mathbb{R}^3$
$$ a\times (b\times c) = (a\cdot c) b - (a\cdot b) c $$
Let $v$ be an arbitrary vector ...
- 35.4k
5
votes
Accepted
A fractional weighted Poincaré inequality
It is not true. Start with a function $u$ which vanishes for $x<0$ and is equal to $1$ for $0 \leq x \leq \frac 12$ and then smooth from $x \geq \frac 12$. The Fourier coefficients behave like $1/n$...
- 4,765
2
votes
Accepted
A question on biharmonic functions
No. Let $w = (1 - |x|^{2 - n})_+$, with $n > 2$.
This is globally Lipschitz. It is harmonic when positive, so therefore also biharmonic. it is subharmonic everywhere (max of two harmonic functions ...
- 664
0
votes
Parabolic Schwarz lemma
You are computing $\Delta_g \operatorname{tr}_g(f^{\ast}h) = g^{i \bar{j}} \partial_i \partial_{\bar{j}} (g^{k\bar{\ell}} h_{\gamma \bar{\delta}} f_k^{\gamma} \bar{f_{\ell}^{\delta}})$ in normal ...
- 1,249
2
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Regularity of solution of $(-\Delta + w)f = 0$
regarding the case $f\in L^n$ I can give you some suggestion.
Since $L^n$ is contained in the Morrey space $L^{1,n-1}$ and you have Laplace operator you can say that
$\nabla f $ belongs to BMO and ...
7
votes
Accepted
Regularity of solution of $(-\Delta + w)f = 0$
As discussed in the comments, I interpret the question as asking about the asymptotics of $f'(r)$, $r\to 0+$, for solutions of
$$
-\frac{d^2f}{dr^2} -\frac{2}{r} \frac{df}{dr} + w(r)f(r) = 0 ; \quad\...
- 20.4k
0
votes
Accepted
Solution of a linear hyperbolic PDE
After some fiddling I found the solution, which I'm posting for future reference.
First perform the transformation
$$
u(x,y) = e^{-k(x+y)} v(x,y)
$$
which essentially sets $k=0$ by giving the simpler ...
- 71
3
votes
A priori estimates to $u_t - \Delta u = u^2$
I figured out a way to show it, basically by separating the space and consider where $A^+ := \{u>0\}$ and $A^- := \{u<0\}$. Then the RHS reads
\begin{equation}
\int u^2 \min(u,c) = \int_{\{u\geq ...
- 159
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