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Questions tagged [elliptic-operators]

For questions about elliptic differential operators.

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Approaches to Atiayh Singer index theorem

Soon i will have to choose a scientific adviser. Mathematicians in my university almost explicitly work on theory of ( partial ) differential equations, which i do not really like. But there is one ...
Max393's user avatar
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Approximation with bounded function

Let $\mathbb{D}$ be the unit disc, and let $B(o,r) \subset B(o,r')$ be two balls contained in $\mathbb{D}$. Assume that we have a $C^{\infty}$ function $f: \mathbb{D} \to [a,b]$ which has all its ...
AMHG's user avatar
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Equivariant Multiplicative Formula for Sphere Bundles in the Index Theorem

I am reading through the proof of the Atiyah-Singer index theorem, out of Lawson-Michelson, and I'm a bit confused about the proof of multiplicative property for indices of sphere bundles, where I ...
Elie Belkin's user avatar
2 votes
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Question about the index of two elliptic operators over a 4-dimensional Riemannian manifold

Let $X$ be a compact Riemannian 4-manifold, $P$ a principal $G$-bundle over $X$, and $\mathfrak{g}$ be its adjoint bundle. Let $\omega$ be a self-dual conneciton on $P$ (i.e. its curvature $\Omega \in ...
user302934's user avatar
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A inequality involving an exponential non linearity of a PDE

Let $f : \mathbb{R} \to \mathbb{R}$ a continuous function and suppose there exists $\alpha_0 > 0$ such that $$ \lim_{|s| \to +\infty} \frac{|f(s)|}{e^{\alpha |s|^{N/(N-1)}}} = 0, \quad \forall \...
Lucas Linhares's user avatar
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1 answer
50 views

Clarification of estimates of $H^1$ norm and $H^{-1}$ norm.

Consider the following Dirichlet Problem for $u' \in H^1(\Omega)$ \begin{equation} \begin{cases} Lu' = f, \qquad \mbox{in } \quad\Omega \qquad (1)\\ u' = 0, \qquad \mbox{on } \quad \partial \Omega, \...
Jason Curran's user avatar
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Relation between characteristic polynomial and elliptic linear operators

I'm taking a course in functional analysis that is using Rudin's book. I want to ask if there is a relationship between the characteristic polynomials and their elliptic linear operators? I understand ...
cheeseboardqueen's user avatar
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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
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23 views

Two Neumann eigenfunctions with parallel gradients along the boundary of a manifold

Let $(M, g)$ be a compact Riemannian $3$-manifold with boundary. Is it possible to exist two independent Neumann eigenfunctions $u, v \in C^{\infty}(M)$ associated to the same eigenvalue $\mu > 0$ ...
Eduardo Longa's user avatar
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1 answer
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Fredholm index of elliptic operator, reference request

Let $E$ be a vector bundle over a smooth manifold $M$ ($M$ may need to be compact and without boundary). Furthermore let$$T:\Gamma(M,E)\to\Gamma(M,E)$$be an elliptic k-th order differential operator. ...
Filippo's user avatar
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inverse problem about scalar multiplication on koblitz curves (or more exactly the secp256k1)

My problem is given $Q=nP$ to find point $P$ given 257 bits long integer $n$ and point $Q$. It’s something possible on other curves but Koblitz curves have extra characteristic and can’t be converted ...
user2284570's user avatar
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1 answer
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Explicit form of Parametrix for 2nd Order Elliptic Linear PDE in Divergence Form

Suppose we are given the following elliptic operator: $$P(u) = -(a^{ij} (x) u(x)_j)_i $$ where $a^{ij}$ is positive, symmetric and bounded (uniformly elliptic) over a smooth bounded domain $\Omega \...
A. L.'s user avatar
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Reference for boundary regularity of Neumann eigenfunctions

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain with piecewise $C^{\infty}$ boundary. I have seen it implicitly used in several results that Neumann eigenfunctions of the Laplacian on $\Omega$ ...
Lawford Hatcher's user avatar
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Evans-Krylov estimates by $L^p$ estimates?

I reading Lemma 17.16 of Chapter 17 of Gilbarg-Trudinger's elliptic PDEs book. The lemma is as follows Let $u\in C^2(\Omega)$ satisfy $F[u]=0$ in $\Omega$ where $F$ is elliptic with respect to $u$. ...
Y.Guo's user avatar
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Ricci Flow: The existence of potential of Curvature

For a compact Riemannian Manifold $(M,g)$ without boundary. $R$ as the scalar curvature. And $d\mu$ is the Riemannian volume form. So we can define the average of the scalar curvature $r:= \frac{\...
mikeqwertyuiop's user avatar
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references for 2nd order elliptic boundary value problems

Let $\Omega \subset \mathbf{R}^2$ open and bounded with smooth boundary $\partial \Omega$. Suppose that we have a second order differential operator $L$ on $\Omega$ with smooth coefficients. I'm ...
snape1234's user avatar
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Do analytic eigenvalue branches of the Robin problem converge?

Consider the Robin Laplacian eigenvalue problem on a bounded domain $\Omega\subseteq \mathbb{R}^2$: $-\Delta u=\lambda u$ with $\partial_{\nu}u+\alpha u=0$ on $\partial\Omega$. It is well known that ...
Lawford Hatcher's user avatar
4 votes
1 answer
112 views

If $f > 0$, $\ker(L)$ contains only constant functions, where $L = - \Delta + \nabla (- \log(f)) \cdot \nabla$ (Villani, Subsec. 7.6)

In subsection(s) 7.5 (and 4.1) of Topics in Optimal Transportation, Cedric Villani states the following (I paraphrase): Take a probability measure $\mu \in \mathcal P_2(\mathbb R^n)$ with finite ...
ViktorStein's user avatar
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Non-divergence form elliptic operator: Positive for certain input.

Demonstrate that for a uniformly elliptic operator $L$ (the matrix is uniformly elliptic), the inequality $L|x|^q > 0$ holds in the set $\mathbb{R}^n \backslash\{0\}$, as long as the exponent $q$ ...
Lonaldin's user avatar
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Kernel and cokernel of homogeneous Dirichlet boundary problem

I am working with Schwarz’ Hodge Decomposition - A Method for Solving Boundary Problems. It is stated that the existence and $H^2$ regularity of the Dirichlet potential imply that the kernel of the ...
user3766553's user avatar
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0 answers
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Solvability of linear elliptic PDE in unbounded domain.

I am looking for references about the unique solvability of a linear elliptic second order pde in $\mathbb{R}^2$. We can assume that we have decaying conditions at infinity and that the operator is ...
Christos's user avatar
2 votes
0 answers
49 views

Generalization of Fredholm alternative

I am reading a paper about the Harmonic Map Flow onto the sphere $\mathbb S^2$ and at some point in the paper the authors reach the following equation $$L_W(\phi) + h = 0, \quad \phi \cdot W = 0 \quad ...
Falcon's user avatar
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1 vote
1 answer
142 views

Sobolev Bound for solution to Dirichlet Problem from knowledge of boundary estimates

Take some bounded domain, $\Omega \in \mathbb{R}^n$, $n \geq 3$. Denote by $\partial \Omega$ the boundary of $\Omega$, which we take to be Lipschitz. Let $L$ be an elliptic operator that satisfies ...
Jason Curran's user avatar
3 votes
1 answer
46 views

Possible convergence of a basis of a symmetric uniformly elliptic operator in $H^2(U)$

I was not able to solve the following homework problem: Let $U$ be a bounded open set with $C^\infty$ boundary in $\mathbb{R}^n$. Let $L$ be a linear, symmetric uniformly elliptic operator. Let $w_k \...
user82261's user avatar
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0 votes
2 answers
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$H^{1}_0(\Omega)$ norm is invariant under orthogonal maps?

How to prove that $H^{1}_0(\Omega)$ norm is invariant under orthogonal maps? That IS, given $u \in H^{1}_0(\Omega)$ and $T \in O(N)$, How to obtain $$ ||u\circ T||_{H^{1}_0(\Omega)} = ||u||_{H^{1}_0(\...
Lucas Linhares's user avatar
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1 answer
51 views

Properties inherited by the inverse function

I'm doing research about some model arising in electromagnetism and there is a function, let us denote it $$\boldsymbol{\lambda} = \mathbf{g}(\nabla u) = \nu(|\nabla u|)\nabla u,$$ where $u:\Omega\...
Juan David Samboní's user avatar
2 votes
1 answer
158 views

elliptic operators on compact manifolds are bounded

I'm trying to undestand when an elliptic operator on a manifold is bounded. Any elliptic operator in a compact space is Fredholm (see for example this question) In this wikipedia article it ...
Marco's user avatar
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2 votes
1 answer
220 views

Does the Laplace-Beltrami operator on the sphere always commute with the spherical harmonics expansion?

The (real) spherical harmonics $Y^m_\ell$, where where $\ell=0,1,2,\dots$ and $m = -\ell, -\ell+1,\dots,\ell-1,\ell$, are a set of eigenfunctions of the Laplace-Betrami operator $\Delta_{\mathbb S^2}$...
Inzinity's user avatar
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4 votes
0 answers
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Family of spin Dirac operators coming from path of metrics

Let $\mathbb{R}\ni t\mapsto g_t$ be a continuous path (in the $C^{\infty}$-topology of $\Gamma(\text{Sym}^2(T^*M))$) of smooth metrics on a compact spin manifold $M$. For each $t\in \mathbb{R}$, we ...
amnesiac's user avatar
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Estimation for the peridynamic non-local elliptic operator.

Consider a kernel $K:\mathbb{R}^N \to \mathbb{R}$ non negative and symmetric; that is, $K(-x) = K(x)$ for any $x \in \mathbb{R}^N \backslash \{0\}$. Moreover, consider $\gamma K \in L^1(\mathbb{R}^N)...
José's user avatar
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3 votes
0 answers
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Linear Elliptic Problems: Are gradient estimates preserved after perturbation?

We start with the linear elliptic PDE $$ -\operatorname{div}(A\nabla u)=f \quad\text{in}\ \Omega,\\ u=0 \quad\text{on}\ \partial\Omega $$ We assume that $\Omega\subset\mathbb{R}^3$ is a smooth domain, ...
Muschkopp's user avatar
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1 answer
297 views

Prove the Laplacian operator is uniformly elliptic

Given $$Lu=a_{ij}(x)D_{ij}u+b_i(x)D_iu+c(x)u$$ a partial differential operator $L$ is (uniformly) elliptic if there exists a constant $\theta>0$ such that $\sum_{i,j=1}^{n}a^{i,j}(x)\xi_{i}\xi_{j} \...
lebong66's user avatar
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0 answers
64 views

Wavefront set of a distribution and elliptic points on a manifold

Let $M$ be a smooth closed manifold, $E$ a Hermitian vector bundle over $M$ and $P$ a pseudodifferential operator. Let $u\in D’(M,E)$ such that $Pu=0$. I want show that $$ WF(u) \subset T_0^{*}M \...
zarathustra's user avatar
1 vote
0 answers
38 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_\...
Lucas Linhares's user avatar
4 votes
0 answers
359 views

Solvability of elliptic PDE with Neumann boundary condition

I'm trying to better understand the theory of second-order elliptic PDEs on a smooth compact region $\Omega \in \mathbb{R}^N$ with boundary conditions. For the Dirichlet case, there seems to be a ...
JotThisDown's user avatar
1 vote
0 answers
24 views

Elliptic Decomposition Theorem

The famous Hodge Decomposition Theorem can be generalized in the context of elliptic operators. The general decomposition theorem can be found in Theorem 5.5 of Spin Geometry of H. Blaine Lawson and M....
FUUNK1000's user avatar
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1 vote
0 answers
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A maximum principle for elliptical operator in $\mathbb{R}^N$

It is well known that on bounded domain, weak and strong maximum principle works fine. However, I'm wondering if is there some version (with more hypothesis maybe) of this result for elliptical ...
Lucas Linhares's user avatar
1 vote
1 answer
191 views

Weak maximum principle for elliptic operators without classical smoothness assumptions

I am currently reading chapter 8 in Gilbarg-Trudinger's Elliptic PDE of the Second order. Let $L$ denote a strictly elliptic operator (with some additional assumptions on the coefficients). $\Omega\...
Quoka's user avatar
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1 answer
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DeRham complex for $ad(P)$-valued forms

Let $M$ be a smooth manifold. It is well known that the exterior differential $d:\Omega^r(M)\to \Omega^r(M)$ determines an elliptic complex. Let $(P,M,\pi)$ be a principal $G$-bundle over an oriented ...
FUUNK1000's user avatar
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1 vote
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On elliptic operators with non-continuous coefficients

Let $E, F \rightarrow M$ be two smooth vector bundles over a closed smooth manifold $M$, and $P: \Gamma(E) \rightarrow \Gamma(F)$ be a linear elliptic differential operator of order 1 with ...
Taras's user avatar
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1 vote
0 answers
57 views

Regularity resultado for elliptic operator

Let $\Omega$ a smooth bounded domain in $\mathbb{R}^N$. Consider the following operator in the divergent form $$ L(u) = \sum_{i,j=1}^{n}( a_{i,j} u_{x_i})_{x_j} + \sum_{i=1}^{n} b_i u_{x_i} + cu. $$ ...
Lucas Linhares's user avatar
1 vote
0 answers
66 views

How to get the regularity of the pde similar to biharmonic equation on convex polyhedroid?

I'v seen the regularity results for biharmonic equation: $$ \left\{\begin{array}{l} \Delta^2 u=f, \quad \text { in } \Omega, \\ \left.u\right|_{\partial \Omega}=\left.\frac{\partial u}{\partial \...
Chandler's user avatar
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0 answers
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Whn a Weighted Sobolev space is Hilbert?

Consider $V : \mathbb{R}^N \rightarrow \mathbb{R}$ and let $$X(\mathbb{R}^N) = \left\{ u \in H^1(\mathbb{R}^N) : \int_{\mathbb{R}^N} Vu^2 < +\infty \right\}.$$ Also consider the application $$ ||u||...
Lucas Linhares's user avatar
1 vote
1 answer
448 views

A question on Hopf Lemma (strong maximum principle)

I'm following Evans here. The statement is as follows: take a subsolution $u\in C^2(U)\cap C^1(\overline{U})$ of a 2nd order elliptic operator $L$ in non-divergence form. Suppose also $U$ satisfies ...
Mr. Brown's user avatar
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165 views

Hessian of a functional

The question may be very basic, but I am a bit confused about the concepts, so it would be nice if you can clarify them for me and/or suggest some good references to fully understand them. I am ...
Jotabeta's user avatar
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2 votes
0 answers
79 views

Formula for the inverse of Laplacian plus constant in a ball\semiball

Let $\lambda_1$ be the first positive eigenvalue of the following problem in the unit semiball $\mathbb{B}_+^n = \{x \in \mathbb{R}^n : \vert x \vert \leq 1 \text{ and } x_n \geq 0 \}$: \begin{cases} \...
Eduardo Longa's user avatar
1 vote
0 answers
71 views

Motivations for the eigenvalue problem of an elliptic operator

I was looking at the chapter 6.5 of Evans’ book about the eigenvalue problem for (anti-)symmetric elliptic operators, and I was wondering what were the motivations for such a problem. I guess there ...
Spida's user avatar
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0 answers
102 views

Uniform elliptic condition

The definition of uniform elliptic condition. Given an $n \times n$-matrix $A(x)$ depending on some $x$ sometimes I see it framed as $$\lambda I \leq A(x) \leq \Lambda I$$ for $\lambda, \Lambda >...
carlos85's user avatar
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1 vote
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Dependence of the parameter while using the Teorema de Lax–Milgram for solving a PDE

In the Book Evans, Partial Differential Equations, 2010, Theorem 3, p. 319, we have that there's an $\gamma \geq 0$ such that for each $\mu \geq \gamma$ and each function $f \in L^2(\Omega)$, there's ...
Lucas Linhares's user avatar
1 vote
0 answers
46 views

Using a version of Ekeland variational principle to minimize a functional

Consider the problem \begin{cases} -\Delta u = |u|^{p-2}u, \Omega \\ \,\,\,\,\,\,\,\,\,u = 0, \partial \Omega, \end{cases} where $\Omega \subset \mathbb{R}^N$ and $2 < p < 2^{*}$. The functional ...
Lucas Linhares's user avatar

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