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Results tagged with ap.analysis-of-pdes
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user 56624
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
3
votes
Accepted
Harmonic functions and monotonic decay
This is obviously not true if you allow general (i.e., varying along a sphere) boundary values. For example, let the boundary values be positive but zero on the hemispheres facing each other. Then $f( …
- 8,992
2
votes
PDE for the probability of Brownian motion staying in an area (reference request)
The easiest way to show this is to check that if $\hat{u}$ is a bounded solution to your boundary value problem, then $\hat{u}(t-s,B_{s\wedge\tau}+x)$ is a martingale, where $\tau$ is the minimum of t …
- 8,992
2
votes
Estimate of minimum of the Poisson integrals corresponding to a convergent Hausdorff sequenc...
No. Take $\Omega$ to be, say, the unit disc $B_1$ in $\mathbb{C}$ with the defining function $\psi(z)=|z|^2-1$. In fact, any sequence of smooth domains $\Omega_j$ such $B_{1-\frac{1}{j}}\subset\Omega_ …
- 8,992
6
votes
Accepted
Is the $n/2$-th heat kernel coefficient topological?
For $n=2$, the answer is given by $\frac{E}{6}$, where $E$ is the Euler characteristic of $M$, see McKean Jr, H. P., & Singer, I. M. (1967). Curvature and the eigenvalues of the Laplacian. Journal of …
- 8,992
4
votes
Domains with discrete Laplace spectrum
This is more of a literature pointer than an answer. In the case of Dirichlet boundary conditions, the answer to the last question is yes: for example, the cross
$$
C:=\{|x_1|\leq 1\}\cup \{x_2\leq 1\ …
- 4,713
1
vote
The continuous dependence of the Green's function on a domain
On the question of continuity, much more is true: the Green's function is continuous under convergence of domains in the sense of Carathéodory, which is the weakest reasonable topology of planar domai …
- 8,992
4
votes
Accepted
Is the Poisson formula valid when the boundary condition is $ L^2 $?
This is clearly false as stated, since a necessary condition is that $g(x)\to g(\xi)$ as $x\to\xi$ a. e. in the sphere, but if $g$ is merely $L^2$, this may well fail for every point.
The convergence …
- 8,992
1
vote
Probabilistic characterization of first Neumann eigenvalue
The fundamental solution for the heat equation
$$
\partial_t h(t,x)=\Delta_x h(t,x),\quad h(0,x)=\delta_y(x)
$$
has the interpretation of the probability density for the location of the Brownian parti …
- 8,992
2
votes
Accepted
Schwartz regularity for the density of a stochastic process
Using the representation in terms of i. i. d. Gaussians $\xi_1,\xi_2,\dots,$
$$
B_t=\sqrt{2}\sum_{n=1}^\infty (-1)^{n+1}\xi_n\frac{\sin \pi \left(n-\frac12\right)t}{\pi \left(n-\frac12\right)},
$$
we …
- 3,669
6
votes
Accepted
What is an "exact solution" to a PDE?
It depends on context. In the physics literature, there is a term "exactly solvable" meaning that a closed form for the solution can be written; it is never used to indicate that the solution exists i …
- 8,992
9
votes
Accepted
Proof of Green's formula for rectifiable Jordan curves
One can circumvent the technical difficulties as follows. Consider a large ball $K$ containing $\Gamma$ and any $p>2$. Given a function $f\in L^p(K)$, we can define its Cauchy transform
$$
\left(\mat …
- 8,992
2
votes
Solving the Poisson equation using a random walk on $\mathbb Z ^d$
If $\varphi(\cdot)$ is a function defined on a finite $\Omega \subset \mathbb{Z}^d$, define
$$
f(x):=\mathbb{E}^x\left(\sum_{t=0}^{\tau-1}\varphi(X_t)\right),
$$
where $\mathbb{E}^x$ means the expecta …
- 8,992
0
votes
existence of a special conformal mapping
You can always find a map $\Phi$ (unique up to shifts) such that $\Im\mathfrak{m}(\Phi(z)-z)$ is bounded, but, in general, $\Re\mathfrak{e}(\Phi(z)-z)$ may be unbounded.
Let $h$ be the bounded harmon …
- 8,992
1
vote
Scaling of distributions
This is wrong. Take, for example, $p_\epsilon(x)=\frac1\epsilon \varphi(x/\epsilon)$ and $\varphi(x)=\varphi_0(x-1)-\varphi_0(x+1)$, where $\varphi_0$ is a smooth cap supported on $(-1,1)$. Then
$$
\i …
- 8,992