Questions tagged [elliptic-operators]

For questions about elliptic differential operators.

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Existence and uniqueness of solutions to an elliptic system of linear 2nd order PDEs with Neumann boundary conditions

I am considering the following elliptic system of PDEs. I would like to establish existence of solutions in the Holder space $C^{2,\alpha}$ for some $0<\alpha<1$. Let $\Omega\subset\mathbb R^n$ ...
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Asymptotics of exterior solution to perturbed Laplace equation

Suppose we have an elliptic operator $\sum_{i,j=1}^na_{ij}(x)D_{ij}$ on $\mathbb R^n$ for $n\ge 3$ which is asymptotic to the Laplacian i.e. there exists $\beta>0$ so that $$|a_{ij}(x)-\delta_{ij}|\...
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On the regularity of an elliptic PDE involving a power of gradient

Consider $\Omega \subset \mathbb{R}^N$ a bounded regular domain and the problem $$ (P)\,\,\,\ \begin{cases} -\Delta u = \lambda u - u^p + |\nabla u|^q, \Omega \\ \,\,\,\,\,\,\,\,\, u = 0, \,\,\,\,\,\,\...
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Relation between degenerate elliptic operators and elliptic operators

A differential operator of second order $f=f(\cdot,u,Du,D^2u):\mathbb{R}^n\times\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^{n\times n}\to \mathbb{R}$ is said to be degenerate elliptic, if $\forall A\...
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Are there any $L^\infty$ bounds on the eigenfunctions of a first order elliptic dfferential operator on $\mathbb{R}^n$ in terms of eigenvalues?

The question is as in the title. All references I searched for only seem to deal with compact manifolds. So I ask for noncompact cases. For a first-order elliptic differential operator with "...
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Green's alternative formula for integration by parts

I'm currently implementing a method to solve usual elliptic problems where the classical form is the following: $$-\text{div}(k\nabla u) + \vec \beta \cdot \nabla u + \gamma u = f$$ Due to an ...
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Estimates of solution of Elliptic PDE with variable coefficients

Does anyone know any reference that treats elliptic pdes with smooth variable coefficients (preferably on infinite domains, e.g, $\mathbb{R}^2$) and gives an estimate of the solution in terms of the ...
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About first eigenvalue of the Laplace operator

Let $\lambda_1$ the first eigenvalue of the Laplace operator with Dirichlet boundary conditions in $\Omega \subset \mathbb{R}^n$ a bounded domain. Consider the set: $$A:=\{u\in C^\infty(\overline{\...
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Partial Differential Equation with boundary dirichlet conditions without solutions

Let $\Omega$ a smooth and bounded domain, $f\in L^2(\Omega)$ and $L$ an uniform elliptic operator in divergence form. Consider the problem: $$(1) \left\{ \begin{array}{ll} Lu=f \hspace{.4cm} \mbox{ ...
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Inequality $\theta\|{-\Delta u}\|_{L^2(U)}^2 \leq (Lu,-\Delta u),$ for elliptic operator

Let $U$ be the bounded smooth open subset of $\Bbb{R}^n$, with $u \in H^2 \cap H^1_0$. Let $L = \sum_{ij} (a_{ij}(x) u_{x^i})_{x^j} + \sum_k b_k(x) u_{x^k} + c(x) u$ be a general linear differential ...
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Elliptic regularity of weak solutions

I am looking for a precise reference for this result (I hope it is right). If $u\in W^{1,2}(\mathbb{R}^n)$ weak solution to $$\Delta u+b(x)\cdot Du+ c(x)u=0$$ where $b\in C^1$ and $c\in C^0$, then $u\...
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Why does the symbol of the Dirac operator have an "i" in it?

Suppose we have a differential operator of order $n$ that maps sections of a bundle $E \rightarrow M$ to sections of a bundle $F \rightarrow M$: $$ D(\sigma) = \sum_{I} \alpha_{I} \frac{\partial^{I}}{\...
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Constant solution for an elliptic equation

Let $U$ be a open and connected subset of $\mathbb{R}^n$ with regular boundary. Consider the fallowing elliptic problem $$ \Delta u + c(x) u = u^3, U\\ \hspace{2cm}u = 0, \partial U, $$ where $c$ may ...
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Reference for $L^p$ estimates

My PDE professor showed the following result: Let $f \in L^{p}(\Omega)$, for $1 < p < \infty$. Also consider $u \in L^{1}_{loc}(\Omega)$ a solution of \begin{align} -\Delta u + a(x)u&= f, \...
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Elliptic lifting theorem of Laplace-Beltrami Operator.

We know that the Laplace-Beltrami Operator is widly used in many areas, whose definition can be found here. I have learned basic knowledge about PDE and Sobolev spaces, but I know little about the ...
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A norm equivalent to the $H_0 ^1$ norm

Let $\Omega$ be open and bounded, $\ h \in L^2(\Omega)\ $,$\ q \in L^{\infty}(\Omega)$, $\ Lu = -\Delta u + q(x)u$. If $\lambda _1(- \Delta + q(x)) >0$, then the expression $(u,v)_q= \int _{\Omega} ...
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Kernel of Laplacian plus a function

It is known that the kernel of the (non-negative) Laplacian operator on a closed manifold consists of constant functions. I would like to ask if some similar phenomena happens for the modified ...
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Elliptic operators on bounded open sets

Let $L$ be a differential operator of order $k$ defined on a bounded open subset $U$ of a Riemannian manifold $M$ that is elliptic when restricted to a smaller open subset $V$ such that $\bar V\...
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Biharmonic problem with boundary conditions on Laplacian

Let the problem, where $\Omega$ is an open set of $\mathbb{R}^3$ and $h_1$ and $h_2$ are regular given functions \begin{equation}\nonumber%\label{eq:Pe}\tag{$P_{\varepsilon}$} \left\{ \begin{array}[c]...
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Given a generator $L$ on a Riemannian manifold, find an SDE associated to it

Consider a Riemannian manifold $M$ with metric $g$ and a second order elliptic operator of the form $$ Gf=g^{i,j}\frac{\partial^2}{\partial x_i\partial x_j}f+b^i\frac{\partial}{\partial x_i}f=\frac{1}{...
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A coordinate-free criterion for ellipticity of a linear differential operator

In Chapter 6 of Warner's Foundations of Differentiable Manifolds and Lie Groups, he develops a self-contained theory of local elliptic operators to establish the Hodge theorem. I got a bit stuck on a ...
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Elliptic PDE existence theorem for $f \in H^{-1}(\Omega)$

Let $\Omega \subset \mathbb{R}^N$ be a bounded open set and consider the elliptic PDE $$ \begin{align} Lu = f \quad &\textrm{in } \Omega, \\ u = 0 \quad &\textrm{on } \partial \Omega, \end{...
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Is the kernel of an elliptic operator closed subspace of $L^2$?

I wonder if the kernel of an elliptic operator is closed in $L^2$ topology? Maybe there exists theorem which states that an uniform limit of solutions of an elliptic equation is also the solution of ...
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Asymptotic of positive solution to elliptic equation

I am reading the paper "Area minimizing hypersurfaces with isolated singularities" by Hardt and Simon (https://eudml.org/doc/152770) and I get stuck on equation 1.9 on page 106. The ...
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All the solutions of the 2-dim PDE $-\Delta u=\lambda e^{u}$ with boundary conditions $u=0$ are like $\log \frac{8b}{(1+\lambda b r^{2})^2}$

It's clear that with the radial change of variables $r=\sqrt{x^{2}+y^{2}}$, the solutions of the form $\log\left(\frac{8b}{(1+\lambda b r^{2})^2}\right)$ solve the equation $-\Delta u=\lambda e^{u}$ ...
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If a operator is closed then the operator coincides with the maximal and minimal operator?

Let $\sigma\in S^m$ with $m>0$. Then $T_{\sigma}:\mathcal{S}\subset L^p\to L^p$ is the pseudo-differential operators asociated with the symbol $\sigma$. By a Proposition, $T_{\sigma}$ is closable. ...
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Analytical solution to Laplace equation for comparison with FEM solution

I'm currently working on my master's thesis, but I'm stuck. Consider the PDE $\nabla\cdot (\sigma\nabla u)=0$ on $B_1\subset \mathbb{R^3}$, where $\sigma:B_1\rightarrow \mathbb{R}_+$ is constant ...
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On a Riemannian manifold which is globally diffeomorphic to $\Bbb R^n$, can we write the Laplace-Beltrami in the prescribed form on the whole space?

I have this one confusion. Let $G$ be a Riemannian manifold which is globally diffeomorphic to $\Bbb R^n$ . Then can we write its Laplace-Beltrami $L$ in the 'usual elliptic form' i.e. $$\sum_{i,j=1}...
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Using regularity theory of Cauchy-Riemann operator in a certain element of $L^q$

Consider the operator $L_S:W^{1,p}_0(\mathbb{R}\times ]0,1[,\mathbb{R}^n)\times W^{1,p}(\mathbb{R}\times ]0,1[,\mathbb{R}^n)\rightarrow L^p(\mathbb{R}\times ]0,1[,\mathbb{R}^{2n})$ defined by $L_SY=\...
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Nonlocal Harnack inequality

Can someone, please help me to understand on how to obtain the last estimate on page 19 of the following paper, which says follows by using Lemma 2.7 there. My main problem in understanding how the ...
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  • 409
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Regularity of higher order elliptic problem on compact smooth manifolds with boundary

I have trouble in finding a source in the literature for the following result: Let $\overline{M}$ be a compact smooth manifold of dimension $n \in \mathbb{N}$ with interior $M$ and non-empty boundary $...
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estimation use Hodge theory

I'm confused by how we get the following estimation from Hodge theory It seems to me that it is using the fundamental inequality of elliptic operators, but I could not see what this operator is.
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Euler operator is uniformly elliptic

Let $f(p)\in C^2(\mathbb{R}^n)$ be a function such that $\left(f_{p_i p_j}(p)\right)$ is positive definite for all $p\in \mathbb{R}^n$. It follows that $$0<\lambda(p)\leq\Lambda(p),$$ where $\...
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Understand elliptic operators

I have trouble understanding the definition of elliptic operator. I know that for an differential operator to be elliptic, we require that its principal symbol being invertible for nonzero elements, ...
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Is Nehari manifold a topological manifold?

Let $X$ be a separable and reflexive Banach space and consider the $C^1$ functional $I:X\to\mathbb{R}$. We define the Nehari manifold corresponding to $I$ by $$ N=\{u\in X:I'(u)u=0\}. $$ Then the set $...
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Are solutions to this PDE bounded uniformly in $\mathbb{R}^n$

Let $u\in \dot{H}^1(\mathbb{R}^n)$ be a weak solution in dimension $n\geq 3$ to the following PDE, $$-\Delta u = \lambda \rho u$$ where $\rho\in L^{n/2}(\mathbb{R}^n)\cap L^{\infty}(\mathbb{R}^n)$ and ...
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Help in the choice of the best coefficients to study the convergence of the finite element method for PDE

I've been given an exercise which goal is to study the convergence of the finite elements method for an elliptic problem in the form $$ \begin{align} -\text{div}(k \nabla u) + \beta \cdot \nabla u + \...
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2 votes
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Finite difference approximation of elliptic operator: divergence form vs. non-divergence form

Consider the principal part of a standard, strongly elliptic linear operator of 2nd order. In divergence form, this is given by $$ \nabla \cdot (a \nabla u), $$ while in non-divergence form, it is ...
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Convergence of solution of PDE as forcing term varies.

Let $L$ be an elliptic operator. We have the PDE $Lf(x; y) = u(x; y)$ on some domain $\Omega$ with Dirichlet boundary condition. Suppose we know that each PDE admits unique solution. $u$ is a sequence ...
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2 votes
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Is the eigenfunction corresponding to the principal eigenvalue positive within $U$ when $a^{ij} \in L^\infty(U)$?

Theorem 2 (Variational principle for the principal eigenvalue) in Section 6.5 (Eigenvalues and Eigenfunctions) in Evans' book Partial Differential Equations (the first edition) says the function $w_1$,...
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Fréchet derivative of a (nonlinear) differential operator

The question may not be too well-posed, but loosely speaking, suppose $L:W^{1,p}(\mathbb R)\to L^p(\mathbb R)$ is a (possibly nonlinear) first order differential differential operator, such that all ...
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1 vote
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Find strictly subharmonic function vanishing at infinity

I am not sure about the term "strictly" subharmonic. What I want is a function $\psi\in C^{\infty}(\mathbb{R}^3)$ with $\Delta\psi>0$ and $\lim\limits_{|x|\rightarrow\infty}\psi(x)=0$. I ...
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Prove Liouville theorem without using mean value property

How can I prove the following Liouville theorem without using the mean value property? If $u$ is harmonic on $\mathbb{R}^n$ and $\int_{\mathbb{R}^n}|D u|^2 dx \leq C$ for some $C > 0$, then $u$ is ...
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Using Lax Milgram lemma and energy estimates on the real line

I just want to check something. I want to use the energy estimates on the real line for an elliptic operator $L$ acting on $L^2(\mathbb{R})$. (The energy estimates are related to the Lax-Mi https://...
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1 answer
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Fredholm operators -- index

Atkinson's theorem states: $T ∈ L(H)$ is a Fredholm operator if and only if T is invertible modulo compact perturbation, i.e. $TS = I + C_{1}$ and $ST = I + C_{2}$ for some bounded operator S and ...
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Topological index in Atiyah Singer

I'm a beginner at Atiyah-Singer index theorem and I've reviewed some results about theorem. Here's some questions. Ive seen the topological index is equal to $$\operatorname{ch}(D) \operatorname{Td}(X)...
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  • 745
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Hölder continuity of gradient

i would to like know what fact in the pde theory is used in the following article , pg 15 "Since $Dw$ is Hölder continuous, we can see this equation as a linear ellip- tic equation with $C^{\...
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3 votes
2 answers
353 views

Inverting the vorticity via Biot-Savart in Navier Stokes

Given the Navier Stokes equation $\partial_t u+u\cdot \nabla u+\nabla p=\nu \Delta u$ in $\mathbb{R}^3$ with $u$ divergence-free, one is often interested in the vorticity $\omega=\text{curl} \ u$. In ...
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2 votes
1 answer
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Two Definition of Elliptic Symbols

There two definition of elliptic symbol. A smooth matrix function $p(x,\xi)$ is a elliptic symbol of order $m\in\mathbb{R}$ if exist a constant $c>0$ such that for all $|\xi|>c$ we have $p(x,\xi)...
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  • 366
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Poincaré inequality and Lax-Milgram lemma on a convex space

Consider the Hilbert space $H_0^1(\mathbb R^n)$ defined as the closure of $C^\infty_c(\mathbb R^n)$ in $H^1(\mathbb R^n)$. I am interested in the following convex subspace: $$\mathscr C \,:= \left\{\,...
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