All Questions
47 questions with no upvoted or accepted answers
5
votes
0
answers
112
views
Fredholm index of degenerate elliptic PDE
We consider the following degenerate elliptic PDE on the unit ball $B\subset \mathbb{R}^n$:
$$
L(u):= -\operatorname{div}(a \, \nabla u) = 0,
$$
where $a\in C^\infty(B;[0,+\infty))$ satisfies $a(0)=0$...
- 363
5
votes
0
answers
201
views
Where to locate $0\in \Omega$ to get $u_{\varepsilon}(0)\neq 0$ where $\Delta u_{\varepsilon} + (\lambda-\varepsilon) u_{\varepsilon} = \frac{1}{|x|}$
Let $\Omega \subset \mathbb{R}^3$ a smooth bounded domain with $0\in \Omega$ and $u_\varepsilon(x)$ the solution to
$$
\Delta u_\varepsilon + (\lambda-\varepsilon) u_\varepsilon = \frac{1}{|x|}\quad \...
- 241
4
votes
0
answers
111
views
Continuity of solutions of Elliptic PDE with respect to parameters
Let $\alpha \in \mathbb{R}$ and $u_\alpha$ satisfy
$$ \Delta u_\alpha+e^{u_\alpha}=\alpha f(x), \ \ \ \ x\in \mathbb{R}^2$$
where $f$ is a fast decaying smooth function.
I would like to know how the ...
4
votes
0
answers
310
views
Pohozaev identity for linear equations
For $-\Delta u =0$, the Pohozaev identity on say $B_1$ says
$$ \int_{S_1} |u_T|^2 \,d\sigma = \int_{S_1} |u_N|^2 \,d\sigma + (n-2) \int_{B_1} |\nabla u|^2 \ dx$$
Here $u_T$ are the tangential ...
- 455
4
votes
0
answers
89
views
How can I can derive an explicit bound for the solution of the poisson's PDE?
i need some help on this question
Let $\Omega$ be an open subset of $\mathbb{R}^{2}$ (say a square) with
$\partial{\Omega} =\Gamma_{1} \cup \Gamma_{2} \cup\Gamma_{3} \cup\Gamma_{4}$. A structure ...
- 41
3
votes
0
answers
65
views
Convex combination of cyclically monotone sets
I want to show the following statement, but I am not sure how.
Proposition(?):
Let $C \in \mathbb{R}^d$ be a compact convex set, and let $u, v : C \to \mathbb{R}$ be smooth convex functions.
Suppose
$$...
- 51
3
votes
0
answers
62
views
Geometric properties of the unique solution of an elliptic BVP involving the Lie derivative of the metric by a vector field
Setting
Let $(M,g)$ be a compact Riemannian manifold with smooth boundary, and let $\nabla$ be its associated Levi-Civita connection. Consider the following formally self-adjoint, second order linear ...
- 808
3
votes
0
answers
103
views
Comparison principle for Elliptic PDE with exponential nonlinearity
Suppose $\varphi$ is a radial (and radially decreasing) solution of
$$\Delta \varphi+e^{\varphi}=0, \ \ \text{on} \ \ r \in (0,R), $$
with $ R>0$, and $\psi$ is a decreasing radial function ...
3
votes
0
answers
110
views
On the relation between ellipticity and Fredholmness as properties of linear PDE's on Fréchet spaces of smooth sections
Let $M$ be a compact manifold equipped with finite rank vector bundles $E$ and $F$ with spaces of $C^{\infty}$ sections denoted $\Gamma(E)$ and $\Gamma(F)$ respectively. It is standard that a ...
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
2
votes
0
answers
238
views
What is the fundamental solution for the backward heat equation?
According to the theorem of Malgrange and Ehrenpreis a fundamental solution exists for any PDE with constant coefficients. But I didn't manage to find in the literature an explicit formula for the ...
- 2,715
2
votes
0
answers
183
views
Regularity of elliptic partial differential equation with mixed Dirichlet-Robin boundary condition, to prove $u\in H^{2}(\Omega)$
I have posted this problem on Math Stackexchange but got no reply.
When I deal with the wave equation with dynamical boundary condition, I am confused by the regularity of the following elliptic ...
2
votes
0
answers
114
views
A maximum principle in $\mathbb{R}^N$
Let $\delta > 0$ and define
$$
H_\delta(x) = \prod_{j=1}^{N} \cosh(\delta x_j), \quad \forall x \in \mathbb{R}^N.
$$
By straightforward calculations we get $\Delta H_{\delta} (x) = \delta^2 H_\...
- 167
2
votes
0
answers
113
views
Solve a coupled PDE in a rectangle
We consider a coupled PDE in a rectangle $\Omega=(-1,1)\times(-1,1)$. For the simplicity, we assume that the functions are periodic in $x_{1}$ direction.
\begin{cases}
\nabla\cdot u=f_{1},\ & \...
- 151
2
votes
0
answers
206
views
Laplacian on a manifold with two boundary components
I am interested in the Laplace equation on knot complements. The full complement of a knot $K$ is in $S^3$, but for compactness, we delete an open tubular neighborhood around $K$. The Laplace PDE on $...
- 217
2
votes
0
answers
169
views
A basic question about the Spectral Theorem
Let $\Omega$ be a bounded open region in $\mathbb{R}^n$ and $\phi_i $ be the eigenfunctions of $-\Delta$ with Dirichlet boundary condition, i.e.
$$-\Delta \phi_i=\lambda_i \phi_i, \ \ \phi_i|_{\...
- 434
2
votes
0
answers
74
views
Bessel decay for nonhomogeneous PDE
I'm interested in the following nonhomogeneous PDE
$$ (\Delta-k^{2})u=-g $$
on the upper-half plane with smooth and integrable Dirichlet boundary condition, where $g$ is a smooth positive function ...
- 53
2
votes
0
answers
106
views
Regularity of Poisson problem with rough coefficients and mixed boundary conditions
Let $d \in \mathbb N$ and $\Omega \subseteq \mathbb{R}^d$ be open, bounded and connected. Let the boundary $\partial \Omega$ be piecewise Lipschitz and partitioned into a Neumann part $\partial \...
- 171
2
votes
0
answers
173
views
Singularity of the solution of a PDE whose coefficients have zeros
The following PDE arises in a problem of finding the stationary measure of a 2d system of stochastic differential equations (see this math.stackexchange post):
$$\mathcal{A}p=0, \quad p\in C^2(\...
- 1,675
2
votes
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 = ...
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_{\...
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
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
1
vote
0
answers
40
views
Is the average of two viscosity sub-solutions to linear elliptic equations is also a sub-solution?
Let $b\in C_b(R;R)$. Consider the following LINEAR equation on $R^2$:
\begin{equation}
u-\partial_{xx}^2 u + (b(x+y)-b(x)) \partial_y u=f\in C^\infty_c(R^2). \tag{1}
\end{equation}
Assume that $...
- 515
1
vote
0
answers
131
views
Regularity of elliptic equation with Neumann boundary conditions
In the context of the regularity of the free boundary of the one-phase (a.k.a. Bernoulli) problem we want to show $C^{1,\alpha}$ regularity of the free boundary implies smoothness of the free boundary ...
- 11
1
vote
0
answers
43
views
Mixed boundary condition of parabolic equations
Let $ \Omega $ be a bounded and smooth domain in $ \mathbb{R}^n $. Assume that
$$
\partial\Omega=\partial\Omega_D\cup\partial\Omega_N,
$$
where $ \partial\Omega_D $ and $ \partial\Omega_N $ are ...
- 1,219
1
vote
0
answers
120
views
Liouville theorem for an elliptic equation with gradient perturbation
How can I prove the following Liouville theorem for an elliptic equation with gradient perturbation?
Let $u \in L^2(\mathbb R^n;\mathbb R)$ be a smooth solution of
$$ -\Delta u + v \cdot \nabla u = 0 ...
- 217
1
vote
0
answers
47
views
Can we find a uniform bound of the solution of a series of linear partial differential equations related to a parameter
Let $\sigma \in[0,1]$,we consider following series of linear partial differential equations related to the parameter $\sigma$,for example
$$
\left\{\begin{aligned}
\Delta \Phi &=\sigma f(x, y) \...
- 809
1
vote
0
answers
126
views
Reference for global theory of Schrödinger operators
Question. What is a good reference to learn about the spectral properties of Schrödinger operators in $\mathbf{R}^n$? I am specifically interested in references that discuss examples where the ...
- 5,048
1
vote
0
answers
70
views
Solutions of constant coefficients differential operator on $\mathbb{R}^n$
Let $D$ be a constant coefficients linear differential operator on complex valued functions on $\mathbb{R}^n$. Let $\tilde D=\mathbb{F}^{-1}\circ D\circ \mathbb{F}$ be its conjugation with the Fourier ...
- 21.8k
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
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
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:
$$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$
$$u = g \,...
- 151
1
vote
0
answers
37
views
Regularity of a flux induced by a potential
Take
$\Omega\subset R^n$ with smooth boundary (take a ball for example)
a function $f\in L^{\infty}(\Omega)$ with support strictly contained in $\Omega$ and with $\int _{\Omega} f \; dx=0$
a scalar ...
- 71
1
vote
0
answers
414
views
Gilbarg-Trudinger's book Theorem 4.13
I am reading Gilbarg-Trudinger's book "Elliptic Partial Differential Equations of Second Order". I do not understand the proof of Theorem 4.13.
Theorem 4.13 is a special case of Kellogg's theorem in ...
- 369
1
vote
0
answers
171
views
Existence of solution?
I am sorry if this question is not at the MO level. But I have not found a reference so I would like ask it here.
Follow this paper :http://www.math.ku.dk/~hugger/articles/CTAC2003.pdf
Let $\mathcal{...
- 131
1
vote
0
answers
99
views
Limit Toward Discontinuous Point of Dirichlet Boundary Value
The question arises from a paper on Schwarz's domain decomposition method (click here).
We consider a bounded domain in $\mathbb{R}^2$ and a curve splits it into two, see the figure below.
Now we ...
- 291
0
votes
0
answers
36
views
Examples of subharmonic functions
Let $A$ be a constant symmetric matrix with $\lambda < A < \Lambda$ and $0<\lambda < \Lambda$ are fixed constants. Let $u$ be a solution of $\text{div}A \nabla u = 0$. Is it true that $\...
- 455
0
votes
0
answers
18
views
Third order estimate for linear elliptic equations
Let $\lambda < A < \Lambda$ be a constant symmetric matrix and $u$ be a $C^{\infty}$(elliptic regularity gives smooth solutions) solution of $\text{div} A \nabla u = 0$. Let $S_1$ be a sphere ...
- 455
0
votes
0
answers
87
views
Curl-Div equation with singular matrix
I want to solve the equation:
$$
\begin{cases}
\nabla \times (A \mathbf v)=f, \quad x\in \Omega \\
\operatorname{div}(\mathbf v)=0,
\end{cases}
$$
where $\Omega \subset\mathbb{R}^n$, is an open set, $...
- 617
0
votes
0
answers
69
views
Inside and up to boundary regularity improvement of linear differential operator
I'm learning elliptic PDEs and a natural question came to me. Consider a constant coefficient linear differential operator defined on the ball $B_r:=\{\sum_{k=1}^n|x_k|^2<r\}$
$$A=\sum a_\alpha\...
- 421
0
votes
0
answers
62
views
Uniqueness problem of constant coefficient differential operator with given boundary information on compact domain
I'm considering the uniqueness problem for a constant coefficient differential operator $A$ on compact domain $\Omega$ with given boundary information such that we have
\begin{equation}\label{...
- 421
0
votes
0
answers
111
views
Reversing heat transfer with respect to time
Fact: One can easily compute heat dispersion in a plane using the heat equation.
Question: Has any research been done on computing the process in the reverse time direction?
That is, given a heat map $...
- 119
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 ...
- 1
0
votes
0
answers
148
views
Singular elliptic PDE: $-h\,\Delta u +\langle \nabla h,\nabla u\rangle =h$
Let $U\subset\mathbb{R}^n$ with $0\in U$. Fix $h\in L^2(U)\cap C^\infty(U)$ with $h(0)=0$. Is there some $C^1$-function $u\neq 0$ in such that $u$ is solution of
$$-h\,\Delta u +\langle \nabla h,\...
- 245
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
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