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.

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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) :=...
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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 ...
G. Blaickner's user avatar
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1 vote
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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 ...
Aidan Backus's user avatar
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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 $\|...
Elio Li's user avatar
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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 $G_{\lambda}(x,p)$ ...
Davidi Cone's user avatar
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 ...
Chris's user avatar
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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 ...
Chris's user avatar
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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 $\dot W^{1,p}(\mathbb{R}^n)$,...
Benjamin's user avatar
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 ...
Elio Li's user avatar
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1 vote
1 answer
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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 ...
Ma Joad's user avatar
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1 vote
1 answer
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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 ...
Partha's user avatar
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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 ...
Elio Li's user avatar
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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 ...
Lorenzo Pompili's user avatar
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} -\...
Paul's user avatar
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2 votes
1 answer
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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 $\...
Elio Li's user avatar
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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 ...
Javier Gargiulo's user avatar
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 ...
user192837465's user avatar
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 ...
Math_Newbie's user avatar
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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|...
sorrymaker's user avatar
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 $\Delta u +e^u=0$...
Elio Li's user avatar
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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 ...
Ilya Kossovskiy's user avatar
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\...
Davidi Cone's user avatar
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 ...
kichr's user avatar
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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.$...
Elio Li's user avatar
  • 719
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 (...
Elio Li's user avatar
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1 answer
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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 } \...
Elio Li's user avatar
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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 ...
MySheperd's user avatar
  • 866
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: ...
Eduardo Longa's user avatar
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 ...
user88544's user avatar
  • 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 $...
user734979's user avatar
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 ...
Davidi Cone's user avatar
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}^...
Keefer Rowan's user avatar
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 ...
Benjamin Pineau's user avatar
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 ...
Davidi Cone's user avatar
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. $\...
G. Blaickner's user avatar
  • 1,137
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 ...
Luis Yanka Annalisc's user avatar
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,...
Luis Yanka Annalisc's user avatar
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 ...
Chris's user avatar
  • 389
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 $ ...
Luis Yanka Annalisc's user avatar
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,...
Naruto's user avatar
  • 83
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 $$ \...
Luis Yanka Annalisc's user avatar
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{\...
sorrymaker's user avatar
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 ...
Leo Moos's user avatar
  • 4,968
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 ...
zelda's user avatar
  • 73
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 $$ \...
Analyst's user avatar
  • 647
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 $x \in \partial D$ ...
sharpe's user avatar
  • 701
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{...
Eduardo Longa's user avatar
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^...
Sean's user avatar
  • 313
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 $c=0$, ...
user486255's user avatar
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. ...
user29999's user avatar
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