All Questions
Tagged with fa.functional-analysis elliptic-pde
230 questions
0
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371
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Harmonic function with Dirichlet boundary condition
Consider the domain $D = \{(x_1, x_2,.., x_n) \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$. Let $D$ be divided into two parts $D_1$ and $D_2$ by the hyperplane $H = \{x_1 = \frac{1}{2}\}$. My question is: ...
- 1
1
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0
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108
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How to define Biharmonic operator for second order sobolev spaces
I am studying an article Link of Article. There author assumes that $\Omega \subset \mathbb{R}^N$, $ N>4 $ . Some where in the paper we have
$$ \Delta^2 (\cdot) - \frac{\lambda}{|x|^4} (\cdot) : W^...
- 371
4
votes
2
answers
505
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Eigenfunctions of the Laplacian on singular spaces
Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian $\Delta$...
- 601
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0
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173
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Regularity of weak solution
I have also posted the question here. Let me explain what difficulties I have. In fact, one may write
\begin{equation}
\partial_1(f-\partial_1 u)=0
\end{equation}
in $\Omega$. Then one may have the ...
CommunityBot
- 1
1
vote
1
answer
390
views
Square Integrable Harmonic Functions in an Infinite Strip
Suppose $S= \left\{x \in \mathbb{R}^3 : a <x_1< b \right\} $ is an infinite strip the three dimensional Euclidean Space.
Is it true that the only $L^2$ harmonic function in this strip is the ...
- 54.2k
0
votes
1
answer
781
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How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?
Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary.
My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that
$\...
CommunityBot
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1
vote
1
answer
732
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Norm equivalent to Sobolev norm? [closed]
On the hyperbolic space $\mathbb{H}^n$, it is known that the spectrum of the Laplacian satisfies $\text{Spec}(-\Delta) \subset [\frac{(n - 1)^2}{4}, \infty)$. Consider the operator $P = -\Delta + a$, ...
- 29.8k
9
votes
1
answer
1k
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Sobolev space for Mixed Dirichlet - Neumann boundary condition
Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...
CommunityBot
- 1
2
votes
0
answers
154
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Asymptotics of "heat" semigroup
Consider a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary. Consider a second order elliptic operator $L$ on $L^2(\Omega)$, defined by either the Dirichlet or Neumann boundary ...
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2
votes
1
answer
340
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Does this linear elliptic equation have a weak solution?
Let $Q = \Omega \times (0,C)$ where $\Omega$ is a bounded domain, write $(x,y) \in Q$ for $x \in \Omega$ and $y \in (0,C)$. Is the problem
$$\Delta_{(x,y)}v = 0\quad\text{in $Q$}$$
$$\frac{\partial v(...
CommunityBot
- 1
3
votes
1
answer
845
views
Moser estimates?
Consider $u$, an $L^2$ solution to the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on a ball $B_1$ of radius 1 centered at $(t_0, x_0)$, say, where $t$ can be treated as a "time" variable. I ...
- 5,169
1
vote
0
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191
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$L^2$ bound on solution of PDE in terms of $L^2$ norm of initial value
Let $u \in H^1((0,T)\times S)$ be the unique solution of
$$u_{tt} + \Delta u =0$$
$$u|_{t=0}= u_0$$
$$u|_{t=T}=0$$
where $u_0 \in H^{\frac 12}(S)$ and $S$ is some Euclidean hypersurface without ...
- 39k
-2
votes
1
answer
193
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Analysis of Sobolev spaces [closed]
I just wanted to know wthether the following is OK or not.
Let $X$ be $H_0^1(\Omega)\bigcap L^{\infty}(\Omega)$, thought of as a subspace of $H^1_0(\Omega)$ and endowed solely with the usual $H^1$ ...
- 902
2
votes
0
answers
553
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Sobolev space for manifold with boundary
For an compact manifold $M$ without boundary, we consider the eigenfunctions $(f_1,f_2,\ldots)$ of some elliptic operator (e.g $\Delta$) with eigenvalue $\lambda_{1},\lambda_{2},\ldots$. To define ...
3
votes
1
answer
266
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$\lVert u\rVert_{W^{2,p}}$ is bounded above by $\lVert \Delta_p u\rVert_{L^2}$ for $u \in W^{1,p}_0 \cap W^{2,p}$?
I work on a bounded domain in $\mathbb{R}^n$ and let $p \geq 2$ and the operator $\Delta_p u = \nabla \cdot (|\nabla u |^{p-2}\nabla u)$.
Does the following inequality (or something similar hold) for ...
- 8,086
3
votes
0
answers
413
views
Continuously dependent on parameters [closed]
How do we check whether the solution is continuouly dependent on parameters?
Let $\Omega$ be a domain with smooth boundary. Say $f$ and $h$ are smooth. Assume that for each $\theta\in (0, 1]$, the ...
- 41
4
votes
1
answer
198
views
Is this inequality true?
Let $\Omega$ be a bounded domain with smooth boundary. Let $$ S=\{u\in C^2(\overline \Omega): \frac{\partial u}{\partial n}=0 \text{ on } \partial\Omega \}.$$ Fix $\Phi\in S$ with $\Phi(x)>0$ for ...
- 52.3k
2
votes
0
answers
146
views
Minimizing some $H^{-1}$ functional over (a subset of) probability densities in $R^d$
Let me consider the following subset of probability measures in $R^d$
$$
\mathcal{K}_M=\left\{0\leq u(x)\in L^1(R^d):\quad \int u(x)dx =1,\,\int|x|^2u(x)dx\leq M,\,\int u(x)|\log u(x)|dx\leq M\right\}
...
CommunityBot
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1
vote
1
answer
260
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A proof of energy functional appearing in the regularity of elliptic and parabolic equations
I have got trapped in this problem for nearly two years when I dealt with regularity of solutions of elliptic and parabolic equations. I have not found a nice proof to support this assertion. Now I am ...
CommunityBot
- 1
4
votes
1
answer
586
views
Upper bounds for the solution of an elliptic PDE depending on a parameter.
Suppose I have the following PDE on $[0,1]^n$
$$\mathcal{L}u = -\nabla \cdot \left(a(x, r)\nabla u\right) = f(x,r), \qquad x\in [0,1]^n,$$
with periodic boundary conditions and $\int f(x) dx =0$ . ...
CommunityBot
- 1
0
votes
1
answer
265
views
find a weak solution in an intersection of Sobolev spaces
In
using-lax-milgram-to-find-a-weak-solution-in-an-intersection-of-sobolev-spaces
the weak solution for
$$
-\Delta^2 u = f \in L^2(U)\\ \\
u|_{\partial U}=\Delta u|_{\partial U} = 0
$$
was discussed,...
CommunityBot
- 1
3
votes
1
answer
344
views
Pseudoinverse of Neumann-Laplacian
Suppose you have the following PDE: find $u \in H^1(\Omega)$ such that
$$-\Delta u = f, \\ \frac{\partial u}{\partial n} = 0. $$
Further assume a solvability condition
$$\int_\Omega f ~\mathrm{d}\...
CommunityBot
- 1
2
votes
0
answers
122
views
A parametrix for the $\bar\partial$ operator adapted to a holomorphic foliation
Let $X$ be a compact complex manifold with (regular) holomorphic foliation given by a holomorphic subbundle $\mathcal F$ of the tangent bundle. The foliation induces a filtration on differential forms....
- 51
5
votes
1
answer
1k
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why is it called the diamagnetic inequality?
How is the Diamagnetic inequality born? Why is it call this name?
Diamagnetic inequality： $\big|\nabla|u|(x)\big|\leq \big|(\nabla+iA)u(x)\big|$.
- 39k
1
vote
2
answers
522
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Variational problems whose lagrangian density depends on derivatives higher than 1.
The usual theory of calculus of variations, as far as I know, is concerned with lagrangian densities which depend on the function and its gradient, namely we try to minimise $\int L(Dw,w,x) dx$. ...
- 27.5k
2
votes
0
answers
242
views
Core of divergence form operator with unbounded coefficient
Consider the unbounded operator $L$ on $L^2(\mathbb{R^d})$ to be the self-adjoint extension of
$$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$ on $C^2_c(\mathbb{R^d})$.
I also assume that $a(x)...
- 617
1
vote
1
answer
1k
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Laplace equation over concentric spheres
Is there a closed formula for the solution of Dirichlet problem ($\Delta u=0$) for annulus $r <|x| < R$, $x \in R^n$ (n>2), with two given boundary value functions, $f$ over $|x|=r$ and $g$ over ...
- 595
5
votes
3
answers
1k
views
Integration by parts for a general negative-definite self-adjoint operator.
I suspect I am asking a very stupid question.
Suppose you have self-adjoint negative-definite operator $L$ densely defined on a space $L^2(\pi)$, with $Lf = \nabla \cdot ( A(x)\nabla f)$, for some ...
- 1,471
3
votes
1
answer
588
views
orthonormal basis of eigenvectors for laplacian on a concave polygon
I am interested in the Laplace operator $\Delta$ on a concave polygon.
When the polygon is convex, it is known that $\Delta: H^2(\Omega) \rightarrow L^2(\Omega)$
is boundedly invertible. In addition, ...
- 3,343
4
votes
2
answers
1k
views
Trace space and Neumann boundary condition
In which sense is it possible to solve $\Delta u=0$, $\partial_\nu u=\phi$, for $\int\phi=0$ on a closed domain, say a ball $B^3\subset\mathbb R^3$?
For example would a $\phi\in L^p(\partial B^3)$, $...
- 52.3k