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Results tagged with ap.analysis-of-pdes
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Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
8
votes
Accepted
Bounded functions satisfying $\Delta u \geq u$ on $\mathbb{R}^n$
No, such a function does not exist. If $u-\Delta u \leq 0$ with $u$ bounded, then $u\leq 0$. In fact, assume that $u>0$ somewhere. If $u$ has a maximum in $x_0$, then $u(x_0)>0$ and $\Delta u(x_0) \le …
- 6,035
5
votes
Accepted
Growth of nonnegative functions satisfying $\Delta u \geq C>0$
Let $a=\sup_\Omega u$ and solve the problem $\Delta v=C$ in $\Omega$ with $v=a$ at the boundary. We get $v(x)= \frac{C}{2d}|x|^2+a-\frac{C}{2d}$. The function $w=u-v$ satisfies $\Delta w \geq 0$ and …
- 6,035
2
votes
Some questions on a paper of Rellich
This is more a long comment than an answer, to show that $L^2$ solutions may exist in some cases. A simple example is $D^2+D+k^2$ in 1d which has solution $e^{-\alpha x}$ with $2\alpha=1+ \sqrt{1-4k^2 …
- 6,035
5
votes
Accepted
Are there $f, g$ such that $\int_{S^{1}} |f'|^{2}+|g'|^{2}d\theta-2\int_{S^{1}}f'g<0,$ where...
No, there are no such functions. Indeed, if $h$ is periodic and has mean zero, then $\|h\|_2 \le \|h'\|_2$. Take $f,g$ as above and write $g=g_1+c$ with $g_1$ having mean zero. Then
$$
\|f'\|_2^2+\|g' …
- 6,035
2
votes
Accepted
On the Schrödinger equation and the eigenvalue problem
Assuming $q>0$ the Schroedinger operator $-\Delta/q$ is associated to the form $a(u,v)=\int_{\mathbb R^n} \nabla u \cdot \nabla v$ in $L^2(\mathbb R^n, q\, dx)$. The form domain consists of all $u\in …
- 6,035
2
votes
Accepted
Gradient estimate and $L^1$ theory for the Laplace operator
One way to get both is to use the estimates $
\|\phi\|_{W^{2,p}(\Omega)} \leq C\|\psi\|_{L^p(\Omega)}$
which hold when $1<p<\infty$ with a constant $C=C(p,\Omega,n)$. Taking $p>n$ by Sobolev embeddin …
- 6,035
5
votes
Accepted
How to use comparison principle to prove the following inequality about Laplace equation?
Let $\psi$ be harmonic in $\Omega$, with $\psi=\phi$ on $\cup _{i \in S} \Gamma_i$, $\psi=m$ on $\cup_{i \not \in S}\Gamma_i$, where $m=\max_{i \not \in S} \max_{\Gamma_i} \phi$. By comparison, $\psi …
- 6,035
3
votes
Reference or proof of a lemma in PDE
Here is a sketch of the proof. By assumption $\int_{B_2} f \Delta \phi=-\int_{B_2} F\cdot D\phi$ for every $\phi \in C_c^\infty (B_2)$. Fix $\eta \in C_c^\infty(B_2)$, $\eta=1$ in $B_1$ and $\psi \in …
- 6,035
2
votes
Decay estimates for simple elliptic equations
The solution decays like $r^{2-n}$ if $n \geq 3$. Setting $u=r^{\frac{1-n}{2}} v$, then
$$v''-\frac{(n-1)(n-3)}{4r^2} v +g(r)v=0$$ with $g=4q$. This gives the result if $n=3$ since $rg(r) \in L^1$ an …
- 6,035
5
votes
Accepted
The behavior of $ \nabla u $ on the boundary for Poisson equations
The first observation is that the $u$ above satisfies $\nabla u=0$ on $\partial \Omega$ if and only if $f$ is orthogonal to all harmonic functions $v$ in $\Omega$, continuous up the the boundary. In f …
- 6,035
4
votes
The solution of Poisson equation and the distance function from the boundary
Let us assume some regularity on $\partial D$ (bounded and $C^2$ suffices). Then the problem above has a unique solution $u \in W^{2,p}(\Omega) \cap W^{1,p}_0(\Omega)$ for every $p<\infty$ and taking …
- 6,035
9
votes
Accepted
Unique solutions to the heat equation on $\mathbb{R}^3$
Sorry, maybe my previous comment was not clear enough. You have $$\frac{d}{dt} \int |u|^2 e^{-|x|}\, dx=-2\int |\nabla u|^2 e^{-|x|}\, dx +2 \int u \nabla u \cdot \frac{x}{|x|}e^{-|x|}\, dx.$$
Now use …
- 6,035
5
votes
Accepted
A detail in one step in a theorem from a paper of Brezis and Merle
This follows from the mean value theorem. Assume that (up to a subsequence) $w_n(x_n) \geq -B$ with $(x_n) \in K$ (a compact subset of $\Omega$). If $x_n \to x_0 \in K$ and $B(x_n,r) \in \Omega$ for e …
- 6,035
1
vote
Accepted
Estimate of the norm of the radial part of a function
Yes, it is true. Take a function $u \in L^2(R^N)$ and expand for $x=r\omega$
$$
u(r\omega)=\sum_{k=0}^\infty u_k(r)P_k(\omega)$$
where $(P_k)$ an orthonormal basis of spherical harmonics in $L^2(S^{N …
- 6,035
1
vote
Accepted
Continuity of solution of a parabolic PDE w.r.t. system parameters
This is only a sketch of an argument that can be used. Assume that $F$ is Lipscthitz and let $y_1,y_2$ be the solutions corresponding to $f_1, f_2$. If $v=y_2-y_1$, then $$|v_t-\Delta v|=|F(f_2,y_2)-F …
- 6,035