Questions tagged [elliptic-pde]
Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
1,063
questions
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20
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Estimate for the operator $A A_D^{-1}$
Let $O\subset\mathbb{R}^d$
be a bounded domain of the class $C^{1,1}$
(or $C^2$
for simplicity). Let the operator $A_D$
be formally given by the differential expression $A=-\mathrm{div}g(x)\nabla$
...
-1
votes
0
answers
59
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 $...
- 99
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votes
0
answers
56
views
Is the solution of the nonlinear elliptic PDE unique for a given energy?
It is well-known that the following semilinear problem
$$ -\Delta u= f(x, u)\text{ in } \Omega,~ u=0 \text{ on } \partial\Omega$$
on a smooth bounded domain $\Omega$ admits infinitely many $H_0^1(\...
- 621
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votes
0
answers
154
views
Bounding the vorticity equation [closed]
Assume there is a solution to the vorticity equation (in component form)
\begin{equation}\label{Eq1}
\dfrac{\partial}{\partial t}v_i + |\textbf{v}|~|\nabla u_i|\cos \alpha_i - |\textbf{u}|~|\nabla v_i|...
- 13
1
vote
0
answers
65
views
$T$ trace, then $Tg(u)=g(T(u))$ for all $u$ on $W^{1,p}$
The trace operator $T$ is defined for bounded domain $U$ with $C^1$ boundaries as the linear, continuous operator
$T: W^{1,p}(U) \rightarrow L^p(\partial U)$
such that
$$
Tu=u\;\text{ on }\partial U
$$...
- 11
1
vote
0
answers
48
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
$$...
- 31
0
votes
1
answer
133
views
Why $-\Delta_g u+\lambda=\lambda \frac{e^{2 u}}{\int_M e^{2 u} d \mu_g}$ type PDE is called 'mean-field equation'?
Why $$
-\Delta_g u+\lambda=\lambda \frac{e^{2 u}}{\int_M e^{2 u} d \mu_g}
$$type PDE is called 'mean-field equation'? It's closely related to moser-trudinger inequality, there are many classical ...
- 607
1
vote
0
answers
96
views
Regularity of elliptic equation with Neumann boundary conditions
In the context of the regularity of the free boundary of the one-phase (a.k.a. Bernoulli) problem we want to show $C^{1,\alpha}$ regularity of the free boundary implies smoothness of the free boundary ...
- 11
2
votes
0
answers
152
views
A question about the regularity of the Schrödinger equation
While reading the article [1], I noticed I don't understand part of the proof of regularity. For the Schrödinger eigenvalue problem,
\begin{cases}
-\Delta u+Vu=\lambda u, &\text{in } \Omega \\
\...
- 31
0
votes
0
answers
24
views
Elliptic operators with Robin boundary conditions
Can it be proved that two elliptic operators with Robin boundary conditions generate an interval $P$-matrix?
$$
-a\Delta u_i = f_i, \quad a\frac{\partial u_i}{\partial n} + bu_i = 0
$$
$$
-a\Delta v_i ...
- 243
4
votes
0
answers
242
views
Does there exist research about equation like $u_{tt}=\det(D_{x}^{2}u)+\dots$?
I have asked this question on Mathematics Stack Exchange yesterday, but there still is no reply.
Does there exist research about equation like $$u_{tt}=\det(D_x^2 u)+\cdots\text{?}$$ That is to say, ...
1
vote
0
answers
69
views
Moser iteration epsilon-regularity for non-linear system in general dimension
I am attempting to prove the following result in general dimension $n$. Given $(M^n,g)$ a Riemannian manifold with $\mathrm{Ric}_g \geq -(n-1)$ and $\mathrm{Vol}_g(B_1(x)) \geq v > 0$ for all $x \...
3
votes
0
answers
70
views
Dirichlet-to-Neumann map is analytic
Let $M^n$, $n \geq 2$, be a compact smooth manifold with boundary and let $I \ni t \mapsto g_t$ be an analytic (with respect to t) $1$-parameter family of Riemannian metrics on $M$. For each $t \in I$,...
- 2,075
2
votes
0
answers
88
views
Dimension of Laplacian eigenspaces along a smooth 1-parameter family of metrics
Let $(M^n,g)$ be a closed Riemannian manifold, $n \geq 2$. For a smooth 1-parameter family $g_t$, $t \in (-\varepsilon, \varepsilon)$, of Riemannian metrics on $M$ with $g_0 = g$, let $\lambda_k(t)$, $...
- 2,075
0
votes
0
answers
28
views
Quantitative global Schauder estimate for solution operator to second order elliptic equation via extension
Let $\Omega\to\mathbb R^n$ be a bounded Lipschitz domain. We can choose an extension operator $E$ such that $E:W^{k,p}(\Omega)\to W^{k,p}(\mathbb R^n)$ is bounded for all $k\ge0$ and $1<p<\infty$...
- 786
5
votes
1
answer
137
views
$L^2$ regularity theory for elliptic equations: Is there another method other that the difference quotient method? Reference request
So i'm interested in the following classical theorem or similar variants.
Consider the following elliptic PDE
$$
-D_\alpha(a^{ij}D_\beta u) = f.
$$
If we assume that the coefficients $a^{ij}$ are ...
- 161
1
vote
0
answers
42
views
Discrete-to-continuum convergence of principal Fokker-Planck eigenvalues
I am looking for a reference justifying the following statement.
Let $L^n$ be any "reasonably consistent" finite-difference approximation of the Fokker-Planck operator in dimension $d=1$
$$
...
- 4,846
4
votes
0
answers
120
views
Trace-class heat semigroups
Let $(M,g)$ be a compact Riemannian manifold and $\Delta_g$ its Laplace operator.
Let $\varphi$ be a test function on $\mathbf{R}_{>0}$. We define the operator on, say, $L^2(M)$
$$T_{\varphi}(u) :=...
user516086
5
votes
0
answers
164
views
Elliptic regularity and Sobolev spaces
Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e.
$$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$
where $a$ are ...
- 1,035
1
vote
0
answers
91
views
Does a gauge-invariant Caccioppoli inequality hold?
(I previously asked this question on Math.SE but got no responses after two weeks.)
Let $V \Subset U$ be domains in a Riemannian manifold $M$, and $W := U \setminus \overline V$. If $u: U \to \mathbb ...
- 430
0
votes
0
answers
116
views
Dirac distribution on a manifold $M$ as a smooth manifold in $C^1(M)^*$, question about its dimension
I have not learned many knowledges on differential geometry, I met this when trying to read the min-max scheme in PDE on manifold, which is in Section3.1.
Let $M_1= \delta_{x_i}$, $x_i \in M$. For $\|...
- 607
0
votes
0
answers
99
views
Existence of Green functions and some properties
Let $\Omega$ be a smooth domain in $\mathbb{R}^N$， $N\geq 3$, $p\in \Omega$ is a fixed point, $\lambda$ is a parameter (can be 0,>0,<0), if there exisits a Green function $G_{\lambda}(x,p)$ ...
- 401
1
vote
1
answer
89
views
Is there literature on the existence of solutions to elliptic systems on unbounded manifolds?
Most of the current literature I've seen is either for compact Riemannian manifolds or unbounded subsets of Euclidean space. In this article, the authors consider a priori bounds on such systems on ...
- 389
3
votes
1
answer
97
views
Zeroth-order term in elliptic estimates
When solving an elliptic equation
$$
Lu = f \ \text{in} \ \Omega
$$
$$
u = 0 \ \text{on} \ \partial \Omega
$$
for an elliptic operator $L$ of order $m$ on a bounded open set $\Omega$, one has the a ...
- 389
1
vote
1
answer
200
views
Estimates for linear elliptic PDE in the whole of $\mathbb{R}^n$
I am struggling to track down any literature on the topic of elliptic regularity when the domain in question is the whole space $\mathbb{R}^n$. Consider the Sobolev spaces $\dot W^{1,p}(\mathbb{R}^n)$,...
- 63
4
votes
2
answers
441
views
Questions about the results about $\Delta u + e^u=0$, $3 \le n \le 9$: no finite Morse index solution, $n \ge 10$: stable radial symmetric solution
I just read the celebrated paper Farina and Dancer, which talks about the following PDE in $\mathbb{R}^n$
$$\Delta u + e^u=0.$$
They proved that when $3 \le n \le 9$, there is no finite Morse index ...
- 607
1
vote
1
answer
176
views
Proving that solutions to elliptic PDE is analytic using Cauchy–Kovalevskaya theorem?
It seems that Cauchy–Kovalevskaya is not commonly used in books on PDE theory. I am thinking about applying it somewhere interesting.
It is known that if $L$ is a uniformly elliptic operator, with ...
- 1,569
1
vote
1
answer
210
views
Moser iteration in dimension $6$
Let $M$ be a closed Riemannian manifold of dimension $6$. We have a function $f\geq 0$ on $M$ satisfying
\begin{align*}
\Delta f \leq gf-\frac{3}{4}f^2
\end{align*}
Where $g$ is another smooth ...
- 749
1
vote
0
answers
84
views
Definition of stable solution of elliptic PDE and the classification of the solution (as the critical points of energy functional)
My questions arise from Here, it seems that I didn't give a clear question, so I rephrase my questions here.
For example, for $$
-\Delta u=f(u) \quad \text { in } \Omega,
$$
we call a solution is ...
- 607
5
votes
0
answers
185
views
Non-uniqueness of solutions to a simple nonlinear elliptic PDE in $\mathbb R^n$
My question is about non-uniqueness of solutions of an elliptic PDE in $\mathbb R^n$ with source term in a scaling-subcritical space (regular, but with too slow decay at infinity), and with some nice ...
- 925
2
votes
1
answer
233
views
Application of Yamabe and Liouville type equation
Let $\Omega$ be a domain in $\mathbb{R}^n$. I am interested in the following critical elliptic partial differential equations (PDEs):
The Yamabe Type Equation (for $n>2$):
\begin{equation}
-\...
- 884
2
votes
1
answer
170
views
A question about a series of solutions to an elliptic PDE in $B_R$ which is compactly convergent as $R \rightarrow +\infty$
My question arises from Here.
I have a series of eigenvalue equations in $B_R$. $$
-\Delta \phi_R+H(x) \phi_R=\lambda_R \phi_R,
$$
where $\lambda_R \geq 0$ is the first nonzero eigenvalue, with $\...
- 607
1
vote
0
answers
68
views
Estimate for the gradient of solutions in an elliptic differential equation in a Sobolev space
Let $\Omega$ be a bounded or unbounded domain in $\mathbf R^{3}$ with a smooth boundary $S$ and a normal vector given by $n$. Now, we consider the following second-order elliptic problem with Neumann ...
2
votes
0
answers
58
views
Doubt on regularity at "Minimal solutions of a semilinear elliptic equations with a dynamical boundary condition"
In the paper Minimal solutions of a semilinear elliptic equations with a dynamical boundary condition by Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami, in Chapter 2 there is a construction of a ...
5
votes
2
answers
378
views
Reconstruction of second-order elliptic operator from spectrum
Let $M$ be a compact smooth manifold, $(\lambda_n)_{n=1}^{\infty}$ be a square-summable monotonically increasing sequence of non-negative numbers, and let $(f_k)_{k=1}^{\infty}$ be continuous ...
- 228
0
votes
0
answers
96
views
Find an explicit solution to $ -u''= u^2\log |u|$
It is known that $u=e^{-|x|^2+\frac{n}{2}}$ is a solution to the equation $-\Delta u= u\log |u|~~ \text{in}~ \mathbb{R}^n$, which can be obtain by a limiting process ($p\to 1^+$) of
$-\Delta u=\frac{u|...
- 621
2
votes
1
answer
112
views
Generalize the conception of 'stable' solution and 'stable outside a compact set' solution of elliptic PDE from $\mathbb{R}^n$ to closed manifold
I want to talk about generalizing the conception of 'stable' solution and 'stable outside the compact set' solution of elliptic PDE from $\mathbb{R}^n$ to closed manifold.
I'm reading $\Delta u +e^u=0$...
- 607
1
vote
1
answer
117
views
What can one say about the Dirichlet problem for Schrödinger equation with negative potential?
Consider the Schrödinger type equation in $\Bbb R^2$:
$$
\Delta f(x,y)+c(x,y)f(x,y)=0
$$
where $c(x,y)$ is a positive (!) function everywhere analytic on the plane, and $\Delta$ is the Laplace ...
1
vote
0
answers
74
views
Existence of Green function for some perturbation of Laplace operator
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$ $(N\geq2)$ and $\lambda>0$ is a small parameter. I wonder if there exists a Green function such that
$$(\Delta+\lambda) G(x,y)=\delta_x\...
- 401
1
vote
0
answers
42
views
How does the constant appearing in a priori estimate for the elliptic Dirichlet problem depend on the domain?
Someone, could you please tell me suitable references on how the constant $C$ appearing in the a priori estimate for the elliptic Dirichlet problem depends on the domain $\Omega$ ?
More precisely, we ...
- 11
3
votes
1
answer
201
views
What is the infinite Morse index solution?
I'm reading the celebrated paper written by Congming Li and Wenxiong Chen, Classification of solutions of some nonlinear elliptic equations, which considered
$$\Delta u = -e^u \ \ in \ \ \mathbb{R}^2.$...
- 607
1
vote
1
answer
122
views
Time varying domain in Chen Wenxiong and Li Congming 's study on $-\Delta u=\exp u$ in $\mathbb R^2$ and $\int_{\mathbb R^2}\exp u(x) \, dx< +\infty$
I'm considering a problem about time varying domain in Chen Wenxiong and Li Congming 's study on $-\Delta u=\exp u$ in $\mathbb R^2$ and $\int_{\mathbb R^2} \exp u(x) \, d x< +\infty$.
LEMMA 1.1 (...
- 607
0
votes
1
answer
129
views
A problem about regularity and mean value property in the Merle and Brezis work on $-\Delta u = V(x) \exp u$ in $\mathbb{R}^2$ plane
I'm reading the Theorem2 in UNIFORM ESTIMATES AND BLOW-UP BEHAVIOR FOR SOLUTIONS OF $-\Delta u=V(x) e^u$ IN TWO DIMENSIONS
They prove that for the solution of
$$
-\Delta u= V(x)\exp u \text { in } \...
- 607
3
votes
0
answers
57
views
Geometric properties of the unique solution of an elliptic BVP involving the Lie derivative of the metric by a vector field
Setting
Let $(M,g)$ be a compact Riemannian manifold with smooth boundary, and let $\nabla$ be its associated Levi-Civita connection. Consider the following formally self-adjoint, second order linear ...
- 866
1
vote
1
answer
69
views
Infimum of the normalized Laplacian eigenvalues
Let $(M^n,g)$ be a compact Riemannian manifold. The spectrum of the Laplacian operator $\Delta_g = -\operatorname{div} \nabla$ consists of an increasing and diverging sequence of positive eigenvalues:
...
- 2,075
1
vote
1
answer
189
views
Harnack inequality for the minimal surface equation
We consider the minimal surface equation $$
(1+|\nabla u|^2) \, \Delta u=\sum_{i,j=1}^n\partial_iu \, \partial_ju \, \partial_{ij}u\quad\hbox{in $B_1\subset\mathbb R^n.$}
$$
If $u\in C^2(B_1)$ is a ...
- 143
8
votes
3
answers
1k
views
Are all positive eigenfunctions principal eigenfunctions?
In a given domain $\Omega$, we have: $\Delta u=-\lambda u$ with $u>0$. Does this mean that $u$ is a principal eigenfunction for $\Delta$ in $\Omega$?
Also, more generally, does this also apply for $...
- 191
2
votes
1
answer
130
views
Strong maximum principle in entire space
Let $n\geq 3$, $K$ is a bounded function in $\mathbb{R}^n$. If $u$ is a nonegative solution but not equal to 0 of $$-\Delta u=Ku^{\frac{n+2}{n-2}}\quad \text{in }\,\mathbb{R}^n.$$
Can we use the ...
- 401
4
votes
1
answer
313
views
Caffarelli-Kohn-Nirenberg-type inequality with nonradial weight
The Caffarelli-Kohn-Nirenberg inequalities are a set of inequalities generalizing the Gagliardo-Nirenberg inequalities and are of the form
$$\||x|^\gamma u\|_{L^p} \leq C\||x|^\alpha \nabla u\|_{L^q}^...
- 327
6
votes
0
answers
179
views
Reference request: Elliptic regularity estimate in domains with $C^{1,\alpha}$ boundary
I'm wondering if there is a reference for the following (or if it's not true). Let $\Omega$ be a bounded domain with $C^{1,\alpha}$ boundary, where $0<\alpha<1$. For the inhomogeneous Dirichlet ...
