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Results tagged with ap.analysis-of-pdes
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user 150653
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
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
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 …
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3
votes
Is this property preserved under weak$^*$ convergence?
Let me do for balls $\Omega_m$ and I use your Edit. Note that $\bar u_m=\frac{1}{|\Omega_m|} \int_{\Omega_m} u_m \to 0$ by H"older inequality and your assumption. Next I use Poincarè-Wirtinger inequa …
- 6,035
3
votes
On a 3D Gagliardo-Nirenberg inequality
This is a special case of embeddings for homogenuous Sobolev spaces and holds if $u \in L^1_{loc}$ with $\nabla u \in L^p$, $1 \leq p<n$ and the usual $p^*$. A proof of this (and much more) is in the …
- 6,035
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
2
votes
Accepted
Approximation on $H^1_0(B)$ and cut-off functions
Do it first for the half-space $\{x_n >0\}=\Sigma$. If $u$ vanishes at the boundary then
$u(x',x_n)^2=2\int_0^{x_n} uD_n u$ and so ($\Sigma_\delta=\{0 <x_n <\delta\}$)
$$
\int_{\Sigma_\delta} |u|^2 \l …
- 6,035
4
votes
Accepted
An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) ...
If $T_tf=g_t*f$ is the heat semigroup you are asking for the norm of $D_{ij}T_t$ from $C^\alpha$ to itself (endowed with the Holder seminorm).
Let $I_\lambda f(x)=f(\lambda x), \lambda >0$. Then $[I_\ …
- 6,035
2
votes
Accepted
Can functions with "big" discontinuities be in $H^1$?
The function $u$ is not in $H^1$ (but you need $\Omega$ to be connected). Assume it is, then $u\wedge 1 =\chi_{\Omega \setminus \omega} \in H^1(\Omega)$ and its gradient is zero a.e. In fact, the grad …
- 6,035
4
votes
Embeddings of the maximal domain for the Laplacian
There is no hope to gain summability without using boundary conditions. For example the function $\frac{1}{z \log z}$ is holomorphic, hence harmonic, and in $L^2$ in the disc (in the complex plane) c …
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5
votes
Accepted
A fractional weighted Poincaré inequality
It is not true. Start with a function $u$ which vanishes for $x<0$ and is equal to $1$ for $0 \leq x \leq \frac 12$ and then smooth from $x \geq \frac 12$. The Fourier coefficients behave like $1/n$ s …
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3
votes
Accepted
Heat equation with nonlocal boundary condition
I keep the reduction of @Andrè Schlicting, assume $|\Omega|=1$ and consider the operator $(I-P)\Delta$ in $L^2_0=\{u \in L^2(\Omega),\ \int_\Omega u=0\}$, where $Pu=\int_{\Omega} u$.
Assuming regular …
- 6,035
2
votes
Accepted
How to show that $ u $ is vanishing in $ \mathbb{R}^3\setminus B_1 $?
A proof that $u=0$ is as follows. Assume that such $u \geq 0 $ is not identically zero. Since $\Delta u \leq 0$, $u$ can never vanish, otherwise it would have an interior minimum. Taking averages on t …
- 6,035
2
votes
Decay estimates for simple elliptic equations
The solution decays like $r^{2-n}$ if $n \geq 3$. Setting $u=r^{\frac{1-n}{2}} v$, then
$$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
4
votes
Regularity of Newtonian potential along smooth boundary
This is a direct proof which gives $V \in C^\infty (\bar \Omega)$ whenever $g \in C^\infty (\bar \Omega)$, $V$ being the Newtonian potential of $g$. As in the proof by @Terry Tao assume that locally $ …
- 6,035
3
votes
Accepted
Boundedness of solutions to a semilinear PDE
Let me give a positive answer perhaps omitting some details.
Fact 1. Let $u'' \geq ku^\alpha$ in $[c,\ell[$ with $k>0, \alpha>1$ and $u,u' \geq 0$. Let $A=u(c)$, then $ \ell \to c$ as $A \to \infty$ …
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