All Questions
Tagged with lp-spaces partial-differential-equations
185
questions
3
votes
1
answer
67
views
Eliminating Neumann boundary condition for elliptic PDE
In his PDE book, Evans demonstrates that for elliptic PDEs with Dirichlet boundary condition, the boundary term can be eliminated:
I am now wondering if this also works with Neumann boundary ...
- 105
1
vote
2
answers
106
views
$L^{\infty}$ (uniform) decay of Dirichlet heat equation $u_t=\Delta u$
Let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^N$. Consider the following initial-boundary value problem for the heat equation:
\begin{equation}
\begin{cases}
u_t=\Delta u\quad\quad\quad\;...
- 795
1
vote
0
answers
40
views
Decay at infinity of $L^2(\mathbb{R}^n)$ functions
I am trying to justify that a (normalized) solution $\phi$ in $L^2(\mathbb{R}^n)$ of:
$-\Delta\phi+f(x)\phi=K\phi$, with $f(x)=0$ in $\Omega$, $f(x)=M$ in $\Omega^c$
has to vanish outside $\Omega$ ...
- 21
0
votes
0
answers
25
views
Dirichlet Problem with $L^p$ Boundary Data
I am seeking a proof of the following result related to the Dirichlet problem with $L^p$ boundary data. I am not quite sure how to approach the proof. Does anyone know where I might find such a proof ...
- 27
2
votes
1
answer
46
views
A property of solution operator of a elliptic PDE involving positive part of a function
For a given $u \in L^2(\mathbb{R}^N)$, there is a unique $S(u) \in H^1(\mathbb{R}^N)$ such that
$$
\int_{\mathbb{R}^N} \nabla S(u) \nabla \varphi + S(u) \varphi = \int_{\mathbb{R}^N} u \varphi, \quad \...
- 1,219
1
vote
0
answers
18
views
Computing $L^2$ norm and quadrature for time-varying solution to PDE?
Setup. On domain $\Omega \times (0,T]$, the parabolic problem
\begin{align}
u_t + \Delta u + u = f
\end{align}
with some appropriate initial and boundary conditions has solution $u(x,t)$, which I ...
- 2,459
2
votes
0
answers
52
views
If $u\in L^1_{\text{loc}}(\Omega)$ and $\Delta u\in L^1_{\text{loc}}(\Omega)$, then $\nabla u \in L^1_{\text{loc}}(\Omega)$?
(And as well, a transcription for those unable to load images:)
Remark 2. There is a local form of Corollary 1, namely if $u\in L^1_{\text{loc}}(\Omega)$ and $\Delta u \in L^1_{\text{loc}}(\Omega)$, ...
- 525
2
votes
0
answers
98
views
$L^p$-estimates for one dimensional wave-equation with lower order pertubation
Suppose $b\in L^\infty(\mathbb{R})$ and $u\colon \mathbb{R}\times \mathbb{R}\to \mathbb{R}$, where $(t,x)\mapsto u(t,x)$ is a solution to $$
\partial_t^2 u = \partial_x^2 u +b(x)\partial_x u,\quad (x,...
- 873
1
vote
0
answers
25
views
Integration by Parts for not so regular Sobolev functions
I am concerned with the following question: Let us assume we have some nice bounded domain $\Omega$
and $u\in W^{2,p}(\Omega)$ for some $1<p<2$. Let us further assume that we know that $-\Delta ...
- 41
2
votes
0
answers
170
views
$W^{1,p}(\Omega)$ estimates of solutions of elliptic equation
Consider the following simple elliptic problem in a bounded domain $\Omega\subset\mathbb{R}^N$:
$$
-\Delta u(x) + u(x) = f(x) \qquad \forall x\in \Omega
$$
with Neumann boundary conditions in $\...
- 41
1
vote
1
answer
51
views
Weak convergence argue
If we take a sequence $\{u_n\}$ that converges weak to u in $L^2(\Omega)$, where $\Omega$ is bounded and $g_n\to g$ weak-* in $L^{\infty}(\Omega)$ then how can I obtain this limit, for all $\varphi \...
- 549
0
votes
0
answers
19
views
How to use an interpolation argument to prove $A\hookrightarrow L^p(\mathbb R)$ for $p\in [2, +\infty]?$
Assume you have that
$$A\hookrightarrow L^2(\mathbb R) \quad\text{ and }\quad B\hookrightarrow L^\infty(\mathbb R).$$
Suppose also that $A\hookrightarrow B$.
By using the above information, is it ...
user603537
1
vote
0
answers
84
views
Asymptotic formula for the $L^2$ norm of a periodic function
Let $\Omega \subset \mathbb{R}^n$ be an open and bounded set and $u^\varepsilon : \mathbb{R}^n \rightarrow \mathbb{R}$ an $\varepsilon$-periodic function, i.e. $$ u(x + \tau) = u(x) \quad \text{for ...
- 1,037
0
votes
0
answers
57
views
Question about convergence in Sobolev spaces
Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$. Is it true that if $(\varphi_n)_{n\geq 1}\subset C^{\infty}_c(\Omega)$ with $\varphi_n\to \...
- 1,912
0
votes
1
answer
58
views
How to prove that $\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\in L^1(\mathbb R^{2n})$?
Let $\Omega$ be an open bounded domain of $\mathbb R^n$. Let $s\in(0, 1)$, $p\in (1, \infty)$ and consider the Banach space
$$X^{s, p}(\Omega)=\{u\in W^{s, p}(\mathbb R^n): u=0 \text{ in } \mathbb R^n\...
- 425