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.
1,092
questions
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, $L^2(M)$
$$T_{\varphi}(u) :=...
5
votes
0
answers
172
views
Elliptic regularity and Sobolev spaces
Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e.
$$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$
where $a$ are ...
1
vote
0
answers
101
views
Does a gauge-invariant Caccioppoli inequality hold?
(I previously asked this question on Math.SE but got no responses after two weeks.)
Let $V \Subset U$ be domains in a Riemannian manifold $M$, and $W := U \setminus \overline V$. If $u: U \to \mathbb ...
0
votes
0
answers
116
views
Dirac distribution on a manifold $M$ as a smooth manifold in $C^1(M)^*$, question about its dimension
I have not learned many knowledges on differential geometry, I met this when trying to read the min-max scheme in PDE on manifold, which is in Section3.1.
Let $M_1= \delta_{x_i}$, $x_i \in M$. For $\|...
0
votes
0
answers
105
views
Existence of Green functions and some properties
Let $\Omega$ be a smooth domain in $\mathbb{R}^N$, $N\geq 3$, $p\in \Omega$ is a fixed point, $\lambda$ is a parameter (can be 0,>0,<0), if there exisits a Green function $G_{\lambda}(x,p)$ ...
1
vote
1
answer
94
views
Is there literature on the existence of solutions to elliptic systems on unbounded manifolds?
Most of the current literature I've seen is either for compact Riemannian manifolds or unbounded subsets of Euclidean space. In this article, the authors consider a priori bounds on such systems on ...
3
votes
1
answer
114
views
Zeroth-order term in elliptic estimates
When solving an elliptic equation
$$
Lu = f \ \text{in} \ \Omega
$$
$$
u = 0 \ \text{on} \ \partial \Omega
$$
for an elliptic operator $L$ of order $m$ on a bounded open set $\Omega$, one has the a ...
1
vote
1
answer
238
views
Estimates for linear elliptic PDE in the whole of $\mathbb{R}^n$
I am struggling to track down any literature on the topic of elliptic regularity when the domain in question is the whole space $\mathbb{R}^n$. Consider the Sobolev spaces $\dot W^{1,p}(\mathbb{R}^n)$,...
4
votes
2
answers
492
views
Questions about the results about $\Delta u + e^u=0$, $3 \le n \le 9$: no finite Morse index solution, $n \ge 10$: stable radial symmetric solution
I just read the celebrated paper Farina and Dancer, which talks about the following PDE in $\mathbb{R}^n$
$$\Delta u + e^u=0.$$
They proved that when $3 \le n \le 9$, there is no finite Morse index ...
1
vote
1
answer
194
views
Proving that solutions to elliptic PDE is analytic using Cauchy–Kovalevskaya theorem?
It seems that Cauchy–Kovalevskaya is not commonly used in books on PDE theory. I am thinking about applying it somewhere interesting.
It is known that if $L$ is a uniformly elliptic operator, with ...
1
vote
1
answer
220
views
Moser iteration in dimension $6$
Let $M$ be a closed Riemannian manifold of dimension $6$. We have a function $f\geq 0$ on $M$ satisfying
\begin{align*}
\Delta f \leq gf-\frac{3}{4}f^2
\end{align*}
Where $g$ is another smooth ...
1
vote
0
answers
86
views
Definition of stable solution of elliptic PDE and the classification of the solution (as the critical points of energy functional)
My questions arise from Here, it seems that I didn't give a clear question, so I rephrase my questions here.
For example, for $$
-\Delta u=f(u) \quad \text { in } \Omega,
$$
we call a solution is ...
5
votes
0
answers
197
views
Non-uniqueness of solutions to a simple nonlinear elliptic PDE in $\mathbb R^n$
My question is about non-uniqueness of solutions of an elliptic PDE in $\mathbb R^n$ with source term in a scaling-subcritical space (regular, but with too slow decay at infinity), and with some nice ...
1
vote
1
answer
256
views
Application of Yamabe and Liouville type equation
Let $\Omega$ be a domain in $\mathbb{R}^n$. I am interested in the following critical elliptic partial differential equations (PDEs):
The Yamabe Type Equation (for $n>2$):
\begin{equation}
-\...
2
votes
1
answer
176
views
A question about a series of solutions to an elliptic PDE in $B_R$ which is compactly convergent as $R \rightarrow +\infty$
My question arises from Here.
I have a series of eigenvalue equations in $B_R$. $$
-\Delta \phi_R+H(x) \phi_R=\lambda_R \phi_R,
$$
where $\lambda_R \geq 0$ is the first nonzero eigenvalue, with $\...
1
vote
0
answers
81
views
Estimate for the gradient of solutions in an elliptic differential equation in a Sobolev space
Let $\Omega$ be a bounded or unbounded domain in $\mathbf R^{3}$ with a smooth boundary $S$ and a normal vector given by $n$. Now, we consider the following second-order elliptic problem with Neumann ...
2
votes
0
answers
60
views
Doubt on regularity at "Minimal solutions of a semilinear elliptic equations with a dynamical boundary condition"
In the paper Minimal solutions of a semilinear elliptic equations with a dynamical boundary condition by Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami, in Chapter 2 there is a construction of a ...
5
votes
2
answers
453
views
Reconstruction of second-order elliptic operator from spectrum
Let $M$ be a compact smooth manifold, $(\lambda_n)_{n=1}^{\infty}$ be a square-summable monotonically increasing sequence of non-negative numbers, and let $(f_k)_{k=1}^{\infty}$ be continuous ...
0
votes
0
answers
96
views
Find an explicit solution to $ -u''= u^2\log |u|$
It is known that $u=e^{-|x|^2+\frac{n}{2}}$ is a solution to the equation $-\Delta u= u\log |u|~~ \text{in}~ \mathbb{R}^n$, which can be obtain by a limiting process ($p\to 1^+$) of
$-\Delta u=\frac{u|...
2
votes
1
answer
113
views
Generalize the conception of 'stable' solution and 'stable outside a compact set' solution of elliptic PDE from $\mathbb{R}^n$ to closed manifold
I want to talk about generalizing the conception of 'stable' solution and 'stable outside the compact set' solution of elliptic PDE from $\mathbb{R}^n$ to closed manifold.
I'm reading $\Delta u +e^u=0$...
1
vote
1
answer
161
views
What can one say about the Dirichlet problem for Schrödinger equation with negative potential?
Consider the Schrödinger type equation in $\Bbb R^2$:
$$
\Delta f(x,y)+c(x,y)f(x,y)=0
$$
where $c(x,y)$ is a positive (!) function everywhere analytic on the plane, and $\Delta$ is the Laplace ...
1
vote
0
answers
83
views
Existence of Green function for some perturbation of Laplace operator
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$ $(N\geq2)$ and $\lambda>0$ is a small parameter. I wonder if there exists a Green function such that
$$(\Delta+\lambda) G(x,y)=\delta_x\...
1
vote
0
answers
45
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 ...
3
votes
1
answer
215
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
$$\Delta u = -e^u \ \ in \ \ \mathbb{R}^2.$...
1
vote
1
answer
128
views
Time varying domain in Chen Wenxiong and Li Congming 's study on $-\Delta u=\exp u$ in $\mathbb R^2$ and $\int_{\mathbb R^2}\exp u(x) \, dx< +\infty$
I'm considering a problem about time varying domain in Chen Wenxiong and Li Congming 's study on $-\Delta u=\exp u$ in $\mathbb R^2$ and $\int_{\mathbb R^2} \exp u(x) \, d x< +\infty$.
LEMMA 1.1 (...
0
votes
1
answer
141
views
A problem about regularity and mean value property in the Merle and Brezis work on $-\Delta u = V(x) \exp u$ in $\mathbb{R}^2$ plane
I'm reading the Theorem2 in UNIFORM ESTIMATES AND BLOW-UP BEHAVIOR FOR SOLUTIONS OF $-\Delta u=V(x) e^u$ IN TWO DIMENSIONS
They prove that for the solution of
$$
-\Delta u= V(x)\exp u \text { in } \...
3
votes
0
answers
59
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 ...
1
vote
1
answer
75
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:
...
1
vote
1
answer
199
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 ...
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 $...
2
votes
1
answer
143
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 ...
4
votes
1
answer
335
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
$$\||x|^\gamma u\|_{L^p} \leq C\||x|^\alpha \nabla u\|_{L^q}^...
6
votes
0
answers
187
views
Reference request: Elliptic regularity estimate in domains with $C^{1,\alpha}$ boundary
I'm wondering if there is a reference for the following (or if it's not true). Let $\Omega$ be a bounded domain with $C^{1,\alpha}$ boundary, where $0<\alpha<1$. For the inhomogeneous Dirichlet ...
1
vote
0
answers
101
views
Brezis-Kato theorem
Let $n\geq 3$, and $u$ satisfies
$$
-\Delta u=K(x)u^{\frac{n+2}{n-2}}
\quad x\in B_1\setminus \{0\},
$$ where
$|K(x)|\leq A$ in $B_1\setminus \{0\}$, and
$u\geq 0$ in $B_1\setminus \{0\}$.
Can we ...
2
votes
0
answers
99
views
Elliptic equations and Fredholms alternative in the non-compact case
Let $M$ be a smooth Riemannian manifold and $E$ be a finite-rank vector bundle over $M$ equipped with a bundle metric $\langle\cdot,\cdot\rangle\in\Gamma^{\infty}(E^{\ast}\otimes E^{\ast})$, i.e. $\...
1
vote
0
answers
38
views
Mixed boundary condition of parabolic equations
Let $ \Omega $ be a bounded and smooth domain in $ \mathbb{R}^n $. Assume that
$$
\partial\Omega=\partial\Omega_D\cup\partial\Omega_N,
$$
where $ \partial\Omega_D $ and $ \partial\Omega_N $ are ...
2
votes
0
answers
65
views
$ \varepsilon $-regularity, harmonic maps vs harmonic heat flow
Let $ \Omega\subset\mathbb{R}^n $ be a bounded domain with smooth boundary and $ (N,h)\subset\mathbb{R}^L $ is a smooth compact Riemannian manifold. Consider the local minimizer $ u\in W^{1,2}(\Omega,...
2
votes
0
answers
99
views
Elliptic estimates in unweighted Sobolev spaces
In several sources (Choquet-Bruhat & Christodoulou 1981, Nirenberg-Walker 1973) estimates for elliptic partial differential equations on a noncompact manifold are derived in weighted Sobolev ...
0
votes
1
answer
91
views
Limit of minimizers of a class of functionals
Assume that $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^n $. Consider a functional
$$
\mathcal{F}(u)=\int_\Omega(|\nabla u|^2+h^{-1}|u-u_0|^2) \, dx
$$
where $ h>0 $ is a parameter and $ ...
3
votes
1
answer
244
views
Minimal surface on $R^3$ with with non Euclidean metric
I am wondering if there is an analog of the following theorem by Morgan and White:
Suppose $g_E$ is the Euclidean metric on $\mathbf R^3$. If $\gamma$ is a closed $C^{k,\alpha}$ curve in $(\mathbf R^3,...
1
vote
1
answer
146
views
How to show that $ u $ is vanishing in $ \mathbb{R}^3\setminus B_1 $?
I come across an interesting question.
Let $ B_r=\{x\in\mathbb{R}^3:|x|\leq r\} $ be the ball in $ \mathbb{R}^3 $ with radius $ r $. Assume that $ u \in C(\mathbb{R}^3\setminus B_1) $ satisfies
$$
\...
4
votes
1
answer
140
views
Continuous up to the boundary without boundary smoothness
Let $\Omega$ be a bounded domain, $f\in L^{\infty}(\Omega),$ and $0\leq u\in H_0^1(\Omega)$ is a non-negative solution of $\Delta u=f$. My question is as follows:
Can we conclude that $u\in C^0(\bar{\...
1
vote
1
answer
149
views
How to find critical points of functionals when there is a boundary?
Banach spaces have a relatively uninteresting topology, because they are contractible. This prevents the direct application of Morse-like min-max arguments to establish the existence of critical ...
6
votes
1
answer
523
views
Nash–Moser–De Giorgi differences
The names of Nash, Moser and De Giorgi are associated to elliptic and parabolic regularity theory.
But what are the differences in the approach between the three contributions? Can you briefly sketch ...
0
votes
1
answer
163
views
An estimate of the gradient of heat kernel
We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R, (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
I have already proved that
$$
\...
2
votes
0
answers
146
views
Regularity of a weak solution to an elliptic PDE with mixed boundary condition
I have a question on the regularity of a weak solution to an elliptic PDE with mixed boundary condition.
Let $\alpha \in (0,1]$ and let $D$ be a bounded $C^{1,\alpha}$-domain. Let $x \in \partial D$ ...
2
votes
1
answer
112
views
Maximizing the first Neumann eigenvalue on disks
Let $D^2$ be a smooth disk and for any Riemannian metric in $D$, let $\mu_1(g)$ be the first positive Neumann eigenvalue of the Laplacian on $(D, g)$. Li and Yau proved that
$$\mu_1(g) \operatorname{...
2
votes
1
answer
161
views
Smooth dependence of parameter of PDE - viscosity solutions
There is a prevalent method called the "Nonlinear adjoint method" in the study of viscosity solution and Hamilton--Jacobi equation, especially equations of the form
$$
u^\varepsilon + H(x,Du^...
0
votes
0
answers
139
views
Generalized harmonic map
Let $M, N$ be closed Riemannian manifolds and $c$ be a constant. For a map $f:M\to N$, define the energy as
$$E(f) = \frac{1}{2} \int_M\Big( \| df(x)\|^2 - c\| f(x) \|^2 \Big) d\mu(x).$$
When $c=0$, ...
2
votes
0
answers
95
views
Strong maximum principle for weak solutions still holds?
By De Giorge, Nash and Moser solutions of
\begin{equation}
\operatorname{div} (A(x) Du) = 0
\end{equation}
where $Du$ denotes the gradient of $u$ and $A$ is a $\lambda,\Lambda$ elliptic matrix. ...