All Questions
97 questions
0
votes
0
answers
117
views
Harnack Inequality for uniformly elliptic PDE via constructing a singularity
I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$, where $A = A(x)$ is uniformly elliptic. The proof outline I am following ...
- 163
2
votes
0
answers
74
views
Bessel decay for nonhomogeneous PDE
I'm interested in the following nonhomogeneous PDE
$$ (\Delta-k^{2})u=-g $$
on the upper-half plane with smooth and integrable Dirichlet boundary condition, where $g$ is a smooth positive function ...
- 53
1
vote
1
answer
234
views
Existence of unique critical points to second order elliptic PDEs
Let $\Omega\subset\mathbb{R}^2$ is bounded and convex and $\partial\Omega$ be smooth. Consider the second order elliptic PDE (1)
$$
\begin{cases}
Lu = f &\text{ on } \Omega\,,\\ u=0 &\text{ ...
- 173
1
vote
2
answers
899
views
Solving the Poisson equation using a random walk on $\mathbb Z ^d$
How do I solve the Poisson equation with the help of a discrete random walk on $\mathbb Z ^d$?
- 8,992
2
votes
1
answer
127
views
Positive form for a homogeneous elliptic pde
I have a pde of the following form:
\begin{align}
&P(x,D)u = f \text{ on } \Omega, \\
&P(x,D) = \sum\limits_{|\alpha|=2m}a_\alpha(x)D^{\alpha},
\end{align}
where one can assume that $f$ ...
- 537
2
votes
0
answers
106
views
Regularity of Poisson problem with rough coefficients and mixed boundary conditions
Let $d \in \mathbb N$ and $\Omega \subseteq \mathbb{R}^d$ be open, bounded and connected. Let the boundary $\partial \Omega$ be piecewise Lipschitz and partitioned into a Neumann part $\partial \...
- 171
0
votes
1
answer
204
views
How do I show continuity of the mixed and weak solution to Zaremba problem?
I am interested in showing continuity/boundedness of the weak solution to the following problem pde:
\begin{align*}
0 &= \mathbf{q} + \mathbf{\nabla}u && \quad x\in \Omega,\\
0 &= \...
CommunityBot
- 1
2
votes
0
answers
173
views
Singularity of the solution of a PDE whose coefficients have zeros
The following PDE arises in a problem of finding the stationary measure of a 2d system of stochastic differential equations (see this math.stackexchange post):
$$\mathcal{A}p=0, \quad p\in C^2(\...
- 1,675
2
votes
3
answers
542
views
BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?
I asked the following question on math.SE (https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d) just over two months ago, and it ...
- 159
3
votes
2
answers
1k
views
Orthogonality to harmonic functions
Let $a_0$ and $b_0$ be smooth compactly supported functions in $B \subset R^3$, $f\in C^1(\Omega)$, and define
$a_n=f\Delta^{-1}(a_{n-1})=-f(x)\int_{B}a_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$
$b_n=f\Delta^{...
0
votes
1
answer
104
views
Poisson Equation across a Hypersurface [closed]
Let $\mathbb{B}(0,1) \subset \mathbb{R}^3$ denote the unit ball. Let $\Gamma = \{x_3=0\}$. Let us assume $f \in L^2(B)$ .Consider the problem
$ \triangle u = f $ in $\mathbb{B}$ in the weak sense such ...
- 17.2k
2
votes
0
answers
683
views
Laplace problem with Robin boundary condition on a wedge
I'm trying to understand what the essential differences between Dirichlet/Neumann and Robin boundary conditions are. Therefore, let $\omega \in \left(0, 2\pi\right)$ and let
\begin{equation*}
\Omega = ...
1
vote
0
answers
71
views
An existence result for solutions of elliptic equations with a mixed boundary problem
Assume that $\Omega$ is a bounded domain such that
$\partial\Omega=\Gamma_1\cup \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are disjoint and closed. Let us consider the following elliptic equations.
...
- 181
3
votes
0
answers
125
views
Partial regularity for transmission problem in corner domains
Let $n=2$ or $3$ and $\Omega \subset \mathbb{R}^n$ be an open bounded domain. Let suppose that $\Omega$ is divided in two subdomains $\Omega_1$ and $\Omega_2$ and we define $\Gamma = \partial \Omega_1 ...
- 143
0
votes
1
answer
152
views
Solution of Poisson equation vanishing at the boundary of any order
Let $f$ be a compactly supported function in $\Omega \subset \mathbb{R}^3$ and
$\Delta u=f$ in $\Omega$
such that $D^{\alpha}u=0$ on $\partial \Omega$ for every multi-index $\alpha$ with $|\alpha| \...
- 91.8k
2
votes
0
answers
102
views
Elliptic equation with Laplace-Beltrami boundary condition
For my research, I've come across the following type of equation (under variational form).
Assume $\Omega\subset\mathbb{R}^d$ is a Lipschitz domain, $\phi \in L^2(\partial \Omega)$ and $\nabla_{\...
- 19.6k
4
votes
0
answers
89
views
How can I can derive an explicit bound for the solution of the poisson's PDE?
i need some help on this question
Let $\Omega$ be an open subset of $\mathbb{R}^{2}$ (say a square) with
$\partial{\Omega} =\Gamma_{1} \cup \Gamma_{2} \cup\Gamma_{3} \cup\Gamma_{4}$. A structure ...
- 325
2
votes
0
answers
178
views
are these norms equivalent?
If it is known that $\sum_{i,j=1}^{n}a_{ij}\xi_i\xi_j\geq \alpha^2|\xi|^2$, where $\xi=(\xi_1,\xi_2,...,\xi_n)\in\mathbb{R}^n$ then can it be said that $\sum_{i,j=1}^{n}a_{ij}\frac{\partial u}{\...
- 157
0
votes
1
answer
247
views
Gradient bounds on Newtonian potentials
Suppose $N \ge 3$ and let $\Phi(x):= C_N |x|^{2-N}$ is the fundamental solution. Let $\Omega$ denote a bounded domain in $ R^N$.
Consider $ -\Delta u(x) = f(x) $ in $\Omega$ with $u=0$ on $ \...
- 201
6
votes
1
answer
608
views
How do solutions of a PDE depend on parameters?
Let $\Omega\subset\mathbb R^n$ be a bounded smooth domain and $\sigma_1,\sigma_2:\Omega\to(c^{-1},c)$ measurable (for some constant $1<c<\infty$).
Let $f\in H^{1/2}(\partial\Omega)=H^1(\Omega)/H^...
- 321
1
vote
0
answers
81
views
About the "method of lines": when are such solutions good approximations for **all** future time?
This question is about approximate solutions to some classes of PDEs obtained using the "method of lines".
For example, for an initial-value problem given by a PDE on a circle, one can choose $n$ ...
- 111
0
votes
1
answer
390
views
Harmonic/Subharmonic lifting of functions on an annulus
Suppose $\Omega_1, \Omega_2 \subset R^2$ are bounded open regions with $\Omega_1 \Subset \Omega_2$. Let $f_1\in C(\partial \Omega_1)$ and $f_2\in C(\partial \Omega_2)$. Is there a function $h\in H^1(\...
1
vote
0
answers
84
views
Potential theory solution for Variable coefficient Poisson with Dirichlet Boundary conditions
I am looking for a potential theory representation for the following equation in $2$D:
$$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$
$$u = g \,...
- 19.6k
0
votes
0
answers
176
views
For a solution of an elliptic equation, if it is 0 on an open subset, then is it 0 identically?
Let $X$ be a compact smooth manifold, $E, F$ be smooth complex vector bundles over $X$, $L$ an elliptic operator between smooth sections of $E$ and of $F$. Suppose $s$ is a section of $E$ such that $...
- 1,906
3
votes
2
answers
272
views
A Global Estimates for Linear Elliptic PDE
Let $\Omega$ be a bounded smooth region in $R^n$ and $u$ satisfy
$-\Delta u+a(x)u=f, \ \ u|_{\partial \Omega}=0$,
where $a(x)\geq 0$ and $f(x)$ are smooth functions. I wonder if the following ...
- 2,494
3
votes
0
answers
399
views
Laplace Equation with Tangential Derivative Prescribed on the Boundary [closed]
I asked this question on MSE. However, I didn't get good answers there so I am seeking for it here. :)
Consider the following Laplace boundary value problem (BVP)
$$\matrix{
{{\nabla ^2}\Phi (x,y)...
CommunityBot
- 1
1
vote
0
answers
37
views
Regularity of a flux induced by a potential
Take
$\Omega\subset R^n$ with smooth boundary (take a ball for example)
a function $f\in L^{\infty}(\Omega)$ with support strictly contained in $\Omega$ and with $\int _{\Omega} f \; dx=0$
a scalar ...
- 71
5
votes
1
answer
363
views
Regularity up to the boundary for the Poisson problem
It seems that the following assertion is widely accepted:
For $k\in\mathbb N$, $p\geq 2$, $\Omega \subset \mathbb R^n$ bounded with $\partial\Omega\in C^{k+2}$ and $f\in W^{k,p}(\Omega)$, the weak ...
- 3,388
2
votes
1
answer
548
views
Does this PDE only have the trivial solution?
Let $(M,g)$ be a closed Einstein manifold of dimension $m>2$ and
$$
\mathrm{Ricc}(g)=\lambda g,
$$
$h$ a symmetric $2$-covariant tensor, $\Delta=\nabla^*\nabla$ the Laplacian on functions as well ...
- 1,500
2
votes
0
answers
114
views
biharmonic equation with L^1 data and Navier Condition
I am reading an article that, a section of it is mentioned below . I have some question about this section. I will ask my question after the section below. I am thanksed if some one could help me , ...
- 371
6
votes
1
answer
1k
views
Gradient estimate for elliptic equation
Given:
1)a bounded domain $\Omega$ in $\mathbb R^n$ of class $\mathcal{C}^{\infty}$
2) the function $f\in L^{\infty}(\Omega)$ with $\int_{\Omega} f=0$
3)$g=(g_i,\ldots,g_n)\in \mathcal{C}^\alpha(\...
- 4,899
1
vote
0
answers
414
views
Gilbarg-Trudinger's book Theorem 4.13
I am reading Gilbarg-Trudinger's book "Elliptic Partial Differential Equations of Second Order". I do not understand the proof of Theorem 4.13.
Theorem 4.13 is a special case of Kellogg's theorem in ...
- 369
8
votes
1
answer
519
views
Mountain Pass theorem for minimization problems with constraints
Let $I[u]$ be a functional on a (possibly infinite dimensional) Hilbert space. Then, under some conditions, the Mountain Pass theorem guarantees the existence of a saddle point (see http://en....
- 60.5k
1
vote
0
answers
171
views
Existence of solution?
I am sorry if this question is not at the MO level. But I have not found a reference so I would like ask it here.
Follow this paper :http://www.math.ku.dk/~hugger/articles/CTAC2003.pdf
Let $\mathcal{...
- 131
8
votes
1
answer
296
views
Failure of Fredholm property of elliptic PDE systems
Roughly speaking, a PDE operator satisfies the Fredholm property if its principal symbol is elliptic and the information provided on the boundary satisfies the Shapiro-Lopatinskii condition.
What can ...
- 13k
0
votes
0
answers
112
views
Estimate for an integral of a function of the solution to a PDE
Let $\Omega \in \mathbb{R}^3$ be a bounded smooth domain. Assume that smooth functions $\sigma_1,\sigma_2$ satisfy $\sigma_1-\sigma_2 \in C_0^\infty(\Omega)$ and
$\lambda\leq \sigma_1, \sigma_2 \leq \...
- 35
1
vote
0
answers
99
views
Limit Toward Discontinuous Point of Dirichlet Boundary Value
The question arises from a paper on Schwarz's domain decomposition method (click here).
We consider a bounded domain in $\mathbb{R}^2$ and a curve splits it into two, see the figure below.
Now we ...
CommunityBot
- 1
3
votes
2
answers
312
views
Let $\mathrm{div}\,(A\,\mathrm{grad}\,u) + b u = f$. Is $(A\,\mathrm{grad}\,u)$ weakly differentiable?
Let us consider the basic linear elliptic PDE
$$
\mathrm{div} (A\,\mathrm{grad}\,u) + bu = f,
$$
with $f\in L^p,$ $A,b$ uniformly bounded. Do we have, for a weak solution $u\in W^{1,p}(\Omega')$,
$$
(...
- 291
0
votes
1
answer
497
views
Harmonic extension in a ball $B(x, r) \subset \mathbb R^n$
I have recently been trying to understand the theory regarding harmonic extensions in $\mathbb R^n$. I have, however, had some difficulties to find the kind of results I am looking for. For that ...
3
votes
1
answer
570
views
Extending a harmonic function in a ball to subharmonic in a larger ball
Consider the Laplace equation in a ball $B(r) \subset \mathbb{R}^n$ of radius $r$:
$$
\begin{cases}
-\Delta u &= 0, \quad \text {in} \quad B(r), \\
\ \ \ \ \ \, u&= g, \quad \text {in}\quad \...
- 4,899
2
votes
1
answer
333
views
Reference request: Boundary behavior and quantitative lower bound for the principal eigenfunction of an elliptic PDE in a ball $B(r)$
Consider the elliptic eigenvalue problem
$$
\begin{cases}
\int_{B(r)} A(x) \nabla u \cdot \nabla \phi \, dx &= \ \ \frac{\lambda_1}{r^2}\int_{B(r)} u \phi \, dx \\
\qquad \qquad \qquad \quad u&...
- 2,494
4
votes
1
answer
586
views
Upper bounds for the solution of an elliptic PDE depending on a parameter.
Suppose I have the following PDE on $[0,1]^n$
$$\mathcal{L}u = -\nabla \cdot \left(a(x, r)\nabla u\right) = f(x,r), \qquad x\in [0,1]^n,$$
with periodic boundary conditions and $\int f(x) dx =0$ . ...
CommunityBot
- 1
3
votes
1
answer
344
views
Pseudoinverse of Neumann-Laplacian
Suppose you have the following PDE: find $u \in H^1(\Omega)$ such that
$$-\Delta u = f, \\ \frac{\partial u}{\partial n} = 0. $$
Further assume a solvability condition
$$\int_\Omega f ~\mathrm{d}\...
CommunityBot
- 1
2
votes
2
answers
153
views
Is the left regularizer for elliptic BVP a left inverse for the principal part?
Take a differential operator with elliptic symbol, consider just the principal part of the operator. Can one invert this principal part with some parametrix type construction (at least construct a ...
CommunityBot
- 1
1
vote
2
answers
106
views
Bound deg 3 partial differential operator on Laplace eigenfunction?
I am no expert on PDE and analysis but I am looking for certain technique from PDE.
Let $D_2$ be the Laplace operator and $f$ is an eigenfunction, i.e., $D_2 f=\lambda f$ for some $\lambda>1$. (or ...
- 34.7k
7
votes
2
answers
905
views
Fredholm alternative result for general elliptic system?
Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading ...
- 3,322
5
votes
1
answer
471
views
Please recommend some literature on the systematical theory of the elliptic systems!
Now I'm interested in the theory of elliptic systems, for example, both the linear and nonlinear case, the exsitence and regularity results, and is there a Fredholm alternative result for the linear ...
- 3,322