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Results tagged with ap.analysis-of-pdes
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user 22815
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
2
votes
0
answers
248
views
long time existence of a nonlinear parabolic equation
I am thinking about a geometric problem which boils down to the following parabolic equation:
Suppose $u=u(r,t)$, $r$ is defined on $[0,1]$ and $t>0$
$$\begin{cases}\displaystyle \frac{\partial u}{\pa …
- 633
6
votes
1
answer
690
views
Derivation of yamabe flow
I am reading papers about yamabe flow. I have a problem about how people derive it as a gradient flow.
Suppose we have $(M,g_0)$, $g(t)=u^{\frac{4}{n-2}}(t)g_0$ is another conformal metric. Let $R=R( …
- 633
1
vote
1
answer
171
views
metric has morse index 2
I am reading Richard Schoen's classical example on the multiplicity of solutions of yamabe problem. He says on $S^1(T)\times S^{n-1}$, there exists a critical number $T_0$ such that if $T\leq T_0$, th …
- 633
7
votes
1
answer
844
views
Reference for Hölder estimate on parabolic equation with Neumann boundary condition
I saw a type of Hölder estimate in Friedman's book: Partial Differential Equations of Parabolic Type (page 200, 3.24) which goes as follows:
Suppose we have a uniformly parabolic equation with Hölder …
- 633
3
votes
0
answers
105
views
Boundedness of Calderon-Zygmund type operator
I am trying to prove the following fact.
Suppose $\varphi\in C_c^\infty(\mathbb{R}^1)$. Define
$$T(u)(x):=P.V.\int_{\mathbb{R}^1}\frac{\varphi(x)-\varphi(y)}{|x-y|^{\frac32}}u(y)dy$$
where P.V. means …
- 633
4
votes
0
answers
169
views
Inequality between Dirichlet and Neumann eigenvalue for Sturm Liouville problem
Consider the following Sturm Liouville problem on an interval $[a,b]$
$$\frac{\mathrm{d}}{\mathrm{d} x}\left[p(x) \frac{\mathrm{d} y}{\mathrm{d} x}\right]+q(x) y=-\lambda w(x) y$$
for given coefficie …
- 633
2
votes
1
answer
184
views
Boardline case of $W^{2,p}$ estimates on elliptic equations
Suppose $u$ is a strong solution of
$$\begin{cases}\Delta u =f &\quad \text{in} \quad B_1(0)\\ u=0 &\quad \text{on}\quad \partial B_1\end{cases}$$
The well known $W^{2,p}$ estimates says if $f\in L^p( …
- 633
4
votes
1
answer
144
views
Boundedness of Riesz potential on Hardy space
I encounter the following claim in one paper:
If $(-\Delta)^{\frac14}u\in L^{2,\infty}(\mathbb{R})$, then $u\in BMO(\mathbb{R})$. Equivalently in its dual version, if $u\in \mathcal{H}^1(\mathbb{R})$ …
- 633
7
votes
0
answers
3k
views
Definition of homogeneous Sobolev spaces
As we know the inhomogeneous Sobolev space (we only consider $s>0$)
$${H}^{s}\left(\mathbb{R}^{n}\right)=\left\{f \in L^2(\mathbb{R}^n):\int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} \x …
- 633