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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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Approximation on $H^1_0(B)$ and cut-off functions

Let $u \in H^1_0(B)$, where $B$ is the unit ball in $\mathbb{R}^N$. Given $\epsilon > 0$, I need to show there exists a function $\chi_\epsilon \in C^\infty_0(\mathbb{R}^N)$ such that $$ \| u - \...
Lucas Linhares's user avatar
3 votes
0 answers
116 views

Existence of very weak solution to the elliptic equation $\partial_i(a^{ij}\partial_j u)=\partial_k\partial_l f$

Let $a^{ij}\in W^{1,n}\cap L^\infty (B^1)$ be uniformly elliptic, i.e. $\lambda|\xi|^2\le a_{ij}(x)\xi_i\xi_j\le \Lambda |\xi|^2$ for a.e. $x\in B^1$, $\xi\in\mathbb R^n$, where $B_1\subset \mathbb R^...
Tian LAN's user avatar
  • 325
2 votes
0 answers
83 views

Elliptic regularity theory in $\mathbb{R}^2$

I recently encountered two papers discussing elliptic PDEs and variational methods. The first paper claims that according to regularity theory, the solution to $-\Delta u = ug(u^2)$ in $\mathbb{R}^2$ ...
sorrymaker's user avatar
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$C^{2+\alpha}$ proprieties

Let $0<\alpha<1$ and $\Omega\subset \mathbb{R}^n$ a $C^{2+\alpha}$ bounded domain. Hi! I am reading the paper: How to appoximate the heat equation with Neumann Boundary Condition by nonLocal ...
Luiza Camile's user avatar
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110 views

How can i show that the last equality in the text is true?

Suppose that $v$ is critical point of $$ f(u)=\frac{1}{2} \int_D|\nabla u|^2-\frac{1}{p+1} \int_D|u|^{p+1}, \quad u \in H_{0, \text{rad}}^1(D),\quad D(r, d)=\left\{z \in R^N: r^2<|z|^2<(r+d)^2\...
Pádua's user avatar
  • 69
6 votes
1 answer
233 views

Elliptic operators over noncompact manifold

We know for two vector bundles $E$ and $F$ over compact manifold $M$,an elliptic operator $D:\Gamma(\mathrm{E})\to \Gamma(\mathrm{F})$ is automatically Fredholm. And for the case $M$ is noncompact, in ...
TaiatLyu's user avatar
  • 325
2 votes
0 answers
74 views

Regularity of solutions to an elliptic boundary value problem

Let $M = [1,\infty)\times S^2$. For an integer $k \geq 2$ and number $\tau<0$, define the space $L^2_{\tau}([1,\infty);H^k(S^2))$ to be all $H^k(S^2)$-valued functions $u$ on $[1,\infty)$ with $\...
Laithy's user avatar
  • 967
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29 views

Elliptic regularity when the parameters depend on one variable

I am looking for references for the following problem. Let $\Omega=B_1\setminus B_{1/2}$, the open unit ball of $\mathbb R^d$ without the closed ball of radius $1/2$. Suppose that $u\in H^1(\Omega)$ ...
username's user avatar
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1 answer
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Derive elliptic maximum principle from weak derivatives

Let $U$ is a connected open set, and $a^{ij}, c^i \in L^\infty (U).$ $a^{ij}$ satisfies the uniform ellipticity condition. Suppose that $u\in H^1(U) \cap C(\overline U)$ satisfies the condition that $$...
Ma Joad's user avatar
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0 answers
17 views

Regularity of nonlinear equation for fractional laplacian

For a bounded smooth domain $\Omega \subset \mathbb{R}^n$, suppose $u\in H_0^1(\Omega)$ is a solution to $$-\Delta u= |u|^{\frac{4}{n-2}}u+u \text{ in } \Omega.$$ one can employ Moser iteration to ...
sorrymaker's user avatar
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1 answer
106 views

Well posedness of the Plateau problem under lack of uniqueness

The title of this question may seem an oxymoron, but let me describe it and you'll see that perhaps it is not. Premises I am analysing the following Plateau problem. Let $G\subsetneq\Bbb R^n$ be a ...
Daniele Tampieri's user avatar
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A question to the proof of Lemma 9 in "Multiple solutions for the Brezis-Nirenberg problem"

I'm currently puzzled by the final portion of the proof for Lemma 9 in the paper "Multiple solutions for the Brezis-Nirenberg problem"(DOI:10.57262/ade/1355867873). In particular, I'm unsure ...
sorrymaker's user avatar
4 votes
0 answers
70 views

Techniques to estimate PDE which are elliptic in some directions and degenerate in others

I am interested in a family of PDE which have defeated my (admittedly rather naive) attempts to prove any regularity or stability estimates. These are systems of PDE which are elliptic in some ...
Aidan Backus's user avatar
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1 answer
321 views

On the weak derivative of $|u|^{(p-2)/2}u$

Let $u$ be a function such that $|u|^{(p-2)/2}u$ is in $H^1_0(G)$, $G$ is open and $p>2$. How can I show that $$ D(|u|^{(p-2)/2}u)=p/2|u|^{(p-2)/2}D(u) \label{1}\tag{1} $$ or how can I show that, ...
Perelman's user avatar
  • 163
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0 answers
90 views

Using a theorem (which is originally set on 2-dim bounded domain in Euclidean space) on a torus

Actually I'm reading a paper on mean-field equation on torus by M.Struwe and G.Tarantello Here, they studied $$\tag{1} -\Delta u=\lambda\left(\frac{e^u}{\int_{\Omega} e^u d x}-\frac{1}{|\Omega|}\right)...
Elio Li's user avatar
  • 729
1 vote
0 answers
33 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 $...
Guohuan Zhao's user avatar
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42 views

Elliptic regularity for Dirichlet problem

Let $\overline{M}=M \cup \partial M$ be a compact manifold with boundary, where $\partial M$ is the boundary of $\overline{M}$ and $M$ is the interior of $\overline{M}$. Let $P$ be an injective ...
user505117's user avatar
1 vote
0 answers
83 views

Any theory on the elliptic operator $Lu=\Delta u + b_iu_i + cu$ when $c>0$

I wonder if there are theories on elliptic operator $$Lu=\Delta u + b_iu_i + cu$$ when $c>0$, when $c<0$, we are glad to have maximum principle, so the bijectivity can be easily analyzed, but I ...
Elio Li's user avatar
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0 answers
52 views

To study the elliptic PDE on complex manifold, when can we treat it as the real case?

I wonder when studying the elliptic PDE on complex manifold, especially studying the existence of solutions, when can we directly study the real case, for example, when studying $$\Delta_c u = f(x,u),$...
Elio Li's user avatar
  • 729
5 votes
1 answer
95 views

Uniqueness of constructed solutions to the Helmholtz equation

My question is regarding the inhomogeneous Helmholtz equation on $\mathbb{R}^3$ with real wavenumber $k$ and outgoing radiation condition \begin{equation} \Delta u + k^2 u = - f \quad \text{and} \quad ...
confused postdoc's user avatar
0 votes
0 answers
37 views

A question about regularity results in the Elliptic case which are given by Schauder theory

I've been reading Jost's lecture notes "Nonlinear Methods in Riemannian and Kählerian Geometry". In section 2.2 he gives a regular results about Elliptic and parabolic equations, but he ...
luyao's user avatar
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2 votes
0 answers
57 views

A question about considering the solution of elliptic PDE with complex Laplacian as the critical point of a functional

I'm considering the elliptic PDE with complex Laplacian, for example, write $$ \Delta_c(\cdot):=-g^{i \bar{j}} \partial_i \partial_{\bar{j}}(\cdot), $$ and $$\Delta_c(u)=f,$$ by [P.Gauduchon, Math.Ann,...
Elio Li's user avatar
  • 729
4 votes
2 answers
345 views

Nontrivial invariant transformations for heat equations

It is well known that if $u$ is a harmonic function on $\mathbb R^2$ then its Kelvin transform defined by $$ v(r,\theta) = u(\frac{1}{r},\theta)$$ is also harmonic for $r>0$. Note that the Kelvin ...
Ali's user avatar
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3 votes
0 answers
61 views

About the naturality of Krasnoselskii genus on Variational Methods

I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
Jacaré's user avatar
  • 131
5 votes
1 answer
260 views

Bochner Laplacian in coordinates

Sorry if this is a too basic question, but I didn't find an answer anywhere: The connection Laplacian, or Bochner Laplacian, is the differential operator acting on $k$-tensor fields $T\in\Gamma^{\...
B.Hueber's user avatar
  • 1,077
0 votes
1 answer
164 views

About the polynomial characterization of $C^{1,\alpha}(\bar{\Omega})$ Hölder space in Lipschitz domain

I have trouble proving the following statement regarding a characterization of $C^{1,\alpha}$: Let $\Omega$ be a Lipschitz domain. $u$ is pointwise $C^{1,\alpha}$ at all points with the same constant $...
Stack_Underflow's user avatar
0 votes
0 answers
72 views

Gradient estimates of linear elliptic PDE

Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain. Assume that $u(x)$ is the classical solution solving $$a_{ij}(x)\partial_{ij}u(x)+b_i(x)\partial_iu(x)+c(x)u(x)=f(x)$$ $$u(x)\Big|_{\...
mnmn1993's user avatar
3 votes
1 answer
107 views

Can gradient zero implies that a function is constant with Hörmander vector fields

Let $X=(X_1,\cdots,X_m)$ be a system of Hörmander vector fields defined on $\mathbb{R}^n$. The Sobolev space $W_{X}^{1,p}(\Omega)$ is defined by $$W_{X}^{1,p}(\Omega):=\{u\in L^p(\Omega)|X_iu\in L^p(\...
Houa's user avatar
  • 561
2 votes
0 answers
175 views

Question about the formula of Green function of Laplacian on sphere

I'm reading a paper which said that the Green function for $\left(-\Delta_g\right)^m$ on $2m$-dimensional closed manifold is of the form $$\tag{1} G_y(x)=\frac{2}{\Lambda_1} \log \frac{1}{d_g(x, y)}+\...
Elio Li's user avatar
  • 729
7 votes
2 answers
388 views

Elliptic regularity on manifolds: Is this true?

Let $(M,g)$ be a Riemannian manifold (without boundary) and denote by $\Delta_{g}$ the Laplace-Beltrami operator (or any other elliptic operator if you wish). I was trying to find a reference for the ...
B.Hueber's user avatar
  • 1,077
0 votes
0 answers
61 views

Some questions about the concept of stable solution of elliptic PDE

For $$ -\Delta u=f(u) \quad \text { in } \Omega, $$ we call a solution is stable if $$ Q_u(\varphi):=\int_{\Omega}|\nabla \varphi|^2 d x-\int_{\Omega} f^{\prime}(u) \varphi^2 d x \geq 0, \quad \forall ...
Elio Li's user avatar
  • 729
1 vote
0 answers
52 views

Proof that, for $u \in H^1$, $\{ u > \alpha \}$ is a quasi open set

I am reading the monograph by A. Henrot, Extremum problems for eigenvalues of elliptic operators. In chapter 2, the notion of a quasi-open set is defined (see the relevant definitions at the end of ...
JZS's user avatar
  • 469
7 votes
2 answers
477 views

Intuition for Agmon-Douglis-Nirenberg ellipticity

First of all, I am sorry if this is a too basic question, but I stumbled over this notion of ellipticity only very recently. I am trying to understand the definition of ellipticity of systems due to ...
G. Blaickner's user avatar
  • 1,209
1 vote
1 answer
137 views

Derivative of Moreau envelope in Hilbert space with respect to regularization parameter $\lambda$ using Hopf-Lax formula?

Let $\mathcal H$ be a Hilbert space, $f \colon \mathcal H \to (- \infty, \infty]$ a proper, convex, lower semi-continuous function and $\lambda > 0$. The $\lambda$-Moreau envelope of $f$ is $$ f_{\...
ViktorStein's user avatar
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0 answers
82 views

Elliptic PDEs in BSDEs and in optimal control

This soft/reference question is related to this MO post of a similar nature. What are some examples of elliptic PDEs appearing in control and BSDEs?
ABIM's user avatar
  • 4,775
1 vote
0 answers
81 views

Regularity of Feynman-Kac formula for a simple diffusion

Let consider the diffusion process given by: $$dX_t = \alpha(X_t) dW_t$$ where $\alpha(x) = \alpha_1\mathbf{1}_{x\geq 0} + \alpha_2\mathbf{1}_{x< 0}$ ($\alpha_1,\alpha_2>0$) and $W$ a Wiener ...
NancyBoy's user avatar
  • 311
0 votes
0 answers
65 views

Examples of symmetry-breaking solitons which retain a subgroup symmetry

There are many works on spontaneous symmetry breaking in the Nonlinear Schrödinger equation with asymmetric soliton solutions. However, all symmetry breaking soliton examples I have seen go from the ...
Leo Anibal's user avatar
5 votes
1 answer
337 views

Elliptic PDEs in Finance

In mathematical finance, one often encounters parabolic PDEs typically through the Feynman-Kac representation theorem/formula. However, I'm curious are there interesting examples of Elliptic boundary ...
ABIM's user avatar
  • 4,775
2 votes
0 answers
69 views

Any solution of an evolution problem tends to a steady state in $L^2$?

I have a general question. Suppose that we have the following simple evolution problem $\begin{cases} \dfrac{\partial u}{\partial t}-\Delta u=f(u), & (t,x)\in (0,\infty)\times\Omega\\ \dfrac{\...
Bogdan's user avatar
  • 1,330
3 votes
0 answers
164 views

$L^{p}$ estimate for $\Delta|\nabla u|$ on a manifold with bounded Ricci curvature

This is a more advanced question than Estimates of $\Delta|\nabla u|$ for harmonic function $u$ . The Bochner formula and refined Kato inequality tells us that on a Riemannian manifold $(M^{n},g)$, if ...
Xin Qian's user avatar
  • 145
0 votes
0 answers
48 views

Non-linearity of viscosity solutions

I am interested in the following problem. Let consider the solution of the non-linear PDE on $[0,T]\times\mathbb{R}$ satifying the following Cauchy problem: $$ \begin{cases} u_t = F(u_{xx}),\\ u(0,x) =...
NancyBoy's user avatar
  • 311
7 votes
1 answer
321 views

Estimates of $\Delta|\nabla u|$ for harmonic function $u$

The well-known Bochner formula and refined Kato inequality (for harmonic function) tells us that for a harmonic function $u$ in $B_{2}\subset\mathbb{R}^{n}$, $$ \frac{1}{n}|\nabla^{2}u|^{2}\leqslant |\...
Xin Qian's user avatar
  • 145
2 votes
0 answers
68 views

Question about the ''crater'' in mountain-pass theorem while reading a paper of solving mean-field equation by mountain-pass theorem

Actually, I'm reading a paper which finds the saddle point of a functional, of course the unbounded below energy functional will suggest a potential saddle, but the structure of mountain pass is the ...
Elio Li's user avatar
  • 729
2 votes
0 answers
39 views

Functional of fully nonlinear equations

Let $\left(\mathcal{M}, g_0\right)$ be a compact Riemannian manifold of dimension $n>2$ and denote by 'Ric' and $R$ respectively the Ricci tensor and the scalar curvature. The $k$-Yamabe problem is ...
Davidi Cone's user avatar
2 votes
1 answer
64 views

Existence of solution to Cauchy boundary value problem in Lipschitz class of functions

For a research question I have run into the following problem that seems intuitively true but I do not know how to prove it and am not sure in which generality. Let $\Omega\subset \mathbb{R}^2$ be a ...
Bobby Nik's user avatar
0 votes
0 answers
133 views

Relative bounds for vorticity

Write the vorticity equation as \begin{equation}\label{Eq20} \begin{split} \dfrac{\partial}{\partial t} v_i & = \biggl[|\textbf{v}|~|\nabla u_i|\cos(\beta_i)- |\textbf{u}|~|\nabla v_i|\cos(\...
MrPie 's user avatar
  • 305
2 votes
1 answer
171 views

Does the strong maximum principle for minimal surfaces hold in Riemannian manifolds?

In Euclidean spaces, the following maximum principle for minimal surfaces are well known. Theorem: If $\Sigma_1$, $\Sigma_2 \subset \mathbb{R}^n$ are complete connected minimal hypersurfaces, $\...
gaoqiang's user avatar
  • 379
2 votes
0 answers
84 views

A question on the maximum principle of second order elliptic equations

Let $Lu=a^{ij}u_{ij} + b^i u_i$ be an elliptic operator of second order in a bounded domain $\Omega$. Assume that $a^{ij}$ is uniformly elliptic. Then it's well known that the following maximum ...
gaoqiang's user avatar
  • 379
3 votes
2 answers
489 views

A problem about how dominated convergence is used in the analysis of variation

I'm reading Existence of solutions to a higher dimensional mean-field equation on manifolds and get stuck on Lemma6. When $\lambda>\Lambda_1$, with $\Lambda_1=(2 m-1) ! \operatorname{vol}\left(S^{2 ...
Elio Li's user avatar
  • 729
4 votes
1 answer
133 views

Embeddings of the maximal domain for the Laplacian

Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain and $n \geq 2$. Consider the subspace of $L^2$-functions whose distributional Laplacian is also an $L^2$-function: $$D = \left\{ f \in L^2(\...
MeS's user avatar
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