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Questions tagged [elliptic-operators]

For questions about elliptic differential operators.

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What should $\tilde{f}$ be in this elliptic equation?

I will quote the following passage from Evans' book- pg 315: Insert $v=(-1)^{|\alpha|}D^{\alpha}\tilde{v}$ into the identity $B[u,v]=(f,v)_{L^2(U)}$. After performing some integrations by parts, we ...
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Estimates on Hessian of Solution to Poisson Equation

Let $\Phi:\mathbb{R}^d\to \mathbb{R}$ be the fundamental solution to the Laplace equation, i.e the unique function $\phi$ such that $\Delta \phi = \delta_0$ in the sense of distributions. The solution ...
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Are Kolmogorov partial differential equations degenerate elliptic?

Let $\mu : \mathbb{R}^d \rightarrow \mathbb{R}^d $ and $\sigma: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d \times d} $ be smooth and Lipschitz continuous. Furthermore, let $\varphi : \mathbb{R}^d \...
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Generalization of Pohozaev identity

Is it possible to generalize Pohozaev identity (and the related non-existence result) to functionals of the form $$\int_\Omega \left(A(x,u,\nabla u) -\frac{\lambda}{2} |u|^{2} - \frac{1}{2^*}|u|^{2^*}...
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Order in a family of elliptic differential operators is constant

In reading the book Spin Geometry by Lawson and Michelsohn I found this definition of a continuous family of elliptic operators: Definition. Let $E,F$ be smooth vector bundles over a smooth ...
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29 views

Stability of an elliptic PDE

I'm reading An introduction to semilinear elliptic equations of Thierry Cazenave. In the middle of the text, he asserts that in general the groundstate solution (that is, the minimal solution with ...
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1answer
53 views

About pseudo-differential operators

Let $\Omega$ be an open and connect subset of $\mathbb{R}^2$,we denote by $\partial \Omega$ its boundary the latter is supposed to be smooth ($\mathcal{C}^\infty)$, its outword normal vector is ...
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Role of Dirac operators in Index Theorems

I'm trying to approach the Atiyah-Singer Index Theorem by getting an overview of the area. One thing that confuses me a lot is that some treatments give (and hence prove) the theorem for Dirac ...
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1answer
34 views

Elliptic Regularity on Convex Domain

In many literature about elliptic regularity on convex domains, they impose that the domain should be polygonal. (In such cases, the inequality below holds) However, it seems to me that the polygonal ...
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Is this statement correct? Local Elliptic Regularity

(The local regularity theorem) Let $P$ be a differential operator of order $k$ that is elliptic over $\bar{U}, U \subseteq \Bbb R^n$ relatively comapct. Let $k,l$ be integers, $f \in W^l$ and $u \in W^...
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eigenfunctions of an globally hypoelliptic operator

An operator $L$ is said globally hypoelliptic in the Schwartz space $\mathcal{S}(\Bbb{R}^{n})$ if $u\in \mathcal{S}'(\Bbb{R}^{n}), Lu\in \mathcal{S}(\Bbb{R}^{n})\Rightarrow u\in \mathcal{S}(\Bbb{R}^{n}...
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Characterization of ellipticity for linear differential operators

Let $P(D)$ be a differential operator of order $m$. I would like to show that $P(D)$ is elliptic if for some open and bounded subset of $\mathbb{R}^n$ and for some real $s$ and $c>0$ and all $\...
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Globally hypoelliptic operator

Are the eigenfunctions of a globally hypoelliptic operator in the Schwarz space $S$.
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Elliptic operator

How to show that the operator \begin{align*} L[u]\equiv \frac {\partial^2 u}{\partial x^2}+\frac {\partial^2 u}{\partial y^2}-\frac {\partial^2 u}{\partial z^2} \end{align*} is not elliptic? Define $...
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Elliptic boundary condition eigenvalue problem

In Chapter 1, section 1.5 of "The Dirac Spectrum" by Nicolas Ginoux, different elliptic boundary conditions for Dirac operators are introduced. On page 24, there is the following theorem Theorem 1....
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decay estimate for a elliptic operator

I have a function $f : \mathbb{R}² \rightarrow \mathbb{C}$ such that $|- \Delta f+ f | < 1/(1+r)^{1+\sigma}$, where $r$ is the distance to the origin and $0<\sigma <1$, and I want to show ...
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operator not satisfying weak maximum principle

I have to prove that the operator $Lu=\Delta u +u$ does not satisfy the weak maximum principle, i.e. if $u$ is such that $Lu=0$, then $u$ can attain its maximum in the interior of the bounded domain $\...
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Is this an hypoelliptic operator?

$$Lu = -\Delta u + \sum\limits_{i = 1}^na_iu(x_i)$$ Is this an elliptic operator? According to definition, it doesn't seem to fit in. Is this an hypoelliptic operator?
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Elliptic regularity of harmonic forms in $L^1$

$\newcommand{\M}{M}$ Let $\M$ be a smooth oriented Riemannian manifold. Let $\sigma$ be a differential $k$-form on $\M$ with coefficients in $L^1(\M)$. We say $\sigma$ is weakly harmonic if $$ \...
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1answer
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A regularity question for elliptic PDE

I revise the question according to the comment of Andrew Let $D$ be an elliptic differential operator with analytic coefficient on $C^{\infty}(\mathbb{R}^2)$, the space of complex valued ...
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Parametrix for an operator on a cylinder

Suppose $T$ is a Dirac-type differential operator on a closed manifold $M$ (or $\mathbb{R}^n$). Let $Q$ be a parametrix for $T$, so that $$QT-1,\qquad TQ-1$$ are with smooth Schwartz kernels. Consider ...
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Reference request: trace class operator

Suppose $D$ is an elliptic differential operator on a non-compact manifold $M$ of order $1$. Then $D$ has a pseudodifferential parametrix $Q$ of order $-1$ such that $$DQ-1=S_0,\qquad QD-1=S_1$$ are ...
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Bound on elliptic operators

I was working on the following problem in PDE (Evans Chapter 6 Problem 5). I already know how to deduce the final statement. What I'm stuck on is in proving that $Lv \leq 0$, for $\lambda$ large ...
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1answer
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Writing differential equation in Elliptic coordinate system

Consider below equation in an ellipsoid: $\nabla^2 f(x,y) = (1 + f(x,y))$ Knowing how to write this equation in polar coordinate with symmetries of a circle: $\partial_r f(r)/r + \partial_r^2 f(r)...
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What does it mean to have a fully nonlinear elliptic PDE?

I am reading chapter 14 of Michael E. Taylor's book, Partial Differential Equations. I am confused because I can't seem to find what it means for a fully nonlinear PDE to be elliptic. Here is the ...
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47 views

Definition of a simple characteristic for an elliptic operator

Could one give a definition of a simple characteristic for an elliptic operator? For example, I have an elliptic operator: \begin{equation} P(x,D) = \sum\limits_{|\alpha|=2m}a_{\alpha}(x)D^{\alpha} +...
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$\;{u|}_{\partial \Omega} =0\;$ implies $\; \frac{\partial u}{\partial τ}=0\;$

I'm reading the following Theorem of boundary estimate for the gradient: If $\;\begin{cases} -\sum_{i,j}a_{ij}u_{x_i x_j}+\sum_{i}b_iu_{x_i}+cu=f\;,\;x\in \Omega \subset \mathbb R^n\\ u=0\;,\...
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1answer
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Question on why a solution of this PDE is of class $\;C^4\;$

Let $\;u:\mathbb R^2 \to \mathbb R^2\;$ ,$\;W \in C^3(\mathbb R^2;\mathbb R)\;$ and consider the following PDE: $\;-\Delta u+W_u(u)=0\;$ where $\;W_u(u)=(\frac{\partial W}{\partial u_1}(u),\...
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1answer
50 views

Counter example to unique solvability of Dirichlet Problem

In Gilbarg and Trudinger, Theorem 6.14 states that for a uniformly elliptic operator $L=A^{ij}D_{ij}+B^iD_i + c$ the Dirichlet problem $ Lu=f $ has a unique solution $u$ for every $f$ under some ...
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Existence and uniqueness of weak solutions to the homogeneous biharmonic equation.

I am currently trying to prove boundedness and coercivity of the bilinear operator $a \colon H^2_0(\Omega) \times H^2_0(\Omega) \to \mathbb{R}$ defined by $$ a(u, v) = \int_\Omega \Delta u \Delta v \,...
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1answer
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Trilaterating 2D cartesian coordinates, without Z

I have a ton of points and distances that I would like to trilaterate. This is not an issue. However, my program takes cartesian coordinates as input and it trilaterates positions X and Y because Z is ...
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Why is the index of a harmonic map finite?

Let $M,N$ be compact Riemannian manifolds, and let $\phi:M \to N$ be harmonic. The index of $\phi$ is defined to be the dimension of the maximal subspaces of $\Gamma(\phi^*TN)$ on which the hessian ...
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1answer
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Definition of constant coefficient elliptic operator

Following are two definitions of constant coefficient elliptic operator: An operator of order m is called elliptic Definition 1: if the top-order symbol $p_m(\xi)$ has no real zeroes except $\xi = 0$. ...
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1answer
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If $A$ generates the $C_0$-semigroup $\{T_t;t\ge0\}$, then $Au=f \Rightarrow u=-\int_0^\infty T_t f dt$?

Let $X$ be a Banach space, $(A,D_A)$ be a linear operator on $X$ which can generate a (contractive, if needed) $C_0$-semigroup $\{T_t;t\ge0\}$, where $D_A\subset X$ is the domain of $A$. The question ...
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Green's functions/fundamental solution to a non-constant coefficients pde

We already know the relationship between Green's function and solution to elliptic partial differential equation, i.e $$u(y)=\int_{\partial \Omega}u\frac{\partial G}{\partial n} ds+\int_\Omega G\Delta ...
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How do I show continuity of mixed-weak solution to Zaremba's problem?

I am interested in showing continuity/boundedness of the solution to the following problem pde: \begin{align*} 0 &= \mathbf{q} + \mathbf{\nabla}u && \quad x\in \Omega,\\ 0 &= \mathbf{\...
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How multiplication operators affect existence and regularity of the solutions of an elliptic PDE

Let $\Omega\subset \mathbb{R}^n$ be a bounded open set. Given $A \in C^\infty( \overline{\Omega},\mathbb{R}^{n\times n})$ and $f\in C^\infty(\overline{\Omega}, \mathbb{R})$ we can search for a weak ...
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1answer
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Why index of elliptic differential operators vanishes in odd dimensional manifolds?

Now I learn the Atiyah-Singer index theorem. When I read an article on wikipedia of the index theorem, I saw an fact "Index of elliptic differential operators vanish in odd dimensional manifolds". I ...
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1answer
244 views

laplace equation with inhomogeneous neumann boundary condition

I want to prove that if the constant $k>0$ is sufficiently large, then for every $f\in L^2(0,1)$, there exists a unigue $u\in H^2(0,1) $ satisfying $$-u''+ku=f \; on \;(0,1) $$ $$u'(0)=0,\;\; u'(1)=...
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Orthogonal Matrix of Maximum Principle

Let $A = ((a^{ij}(x_{0})))$ be a symmetric and positive definite matrix with the size of $n \times n$. Thus, there exists an orthogonal matrix $O=((o_{ij}))$ such that $OAO^{T}=\text{diag}(d_{1},...,...
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Domain of the adjoint of an elliptic operator

I'm just trying to understand the concept of the adjoint of unbounded operators. Let's look at the operator $Au:=-\epsilon\Delta u +(b\cdot\nabla)u+cu$ as an unbounded Operator on $L^{2}(\Omega)$ ...
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Is strongly elliptic operator necessarily properly elliptic?

Let us consider $$ A(D)u:=\sum_{|p|\leq l} a_p D^p u, $$ where $a_p$ may be complex, and define $$ A_0(\xi)= \sum_{|p|=l} a_p \xi^p $$ Def 1The operator $A$ is called strongly elliptic if $$ ...
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Pointwise upper bound on a harmonic function away from boundary

Let $\Omega\subseteq \mathbb{R}^n$ be a domain and let $u$ be a harmonic function on $\Omega$ such that the gradient of $u$ squared integrates to $1$. How can we obtain pointwise upper bounds on $u(x)$...
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Elliptic regularity at boundary

Suppose I have a (non-smooth) domain $\Omega$ on which I have a $H^1$ solution $u$ of a constant coefficient elliptic PDE $L$. Suppose also that $\Gamma$ is a smooth portion of the boundary $\partial\...
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1answer
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Questions on Nonlinear Elliptic Theory by Schauder

I recently started to study about Elliptic theory and below is a brief introduction my professor made: Let $\;u:\mathbb R^n \to \mathbb R\;$ and $\;f:\mathbb R \to \mathbb R\;$ two functions ...
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In what way is $\sigma(L) \in K(T^*X)$ in the $K$-theoretic formulation of the Atiyah-Singer index theorem?

I am asking whether my interpretation of the $K$-theoretic description of the Atiyah-Singer index theorem is correct. First I state what I mean by "$K$-theoretic description". Let $X$ be a manifold ...
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1answer
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Uniform or Holder estimates for positive Schrodinger operators

I am sure this is treated in many standard references, I just can't find them! Let $V(x)$ be a smooth function on $\mathbb{R}^n$, with $V(x)\geq \epsilon>0$ for all $x$. Impose other conditions (...
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The adjoint operators as elliptic operators

Assume that $M$ is a smooth $n$ dimensional manifold with $n>1$. Is there a lie algebra structure on $\chi^{\infty} (M)$, the space of all smooth vector fields on $M$, such that ...
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Convergence rate of finite element method?

Suppose I have a convex polygon $\Omega$ with $n$ sides and I want to find eigenvalues of the Laplacian with mixed Robin boundary conditions: $$ -\Delta u=k^2u\text{ in }\Omega $$ $$ a_iu+b_i\frac{\...
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Laplace transform of $\sum_{i,j=1}^n a_{ij}\frac {\partial ^2 u}{\partial x_i^2 \partial x_j^2}$

It's said that if I have the elliptic partial differential equations: $\sum_{i,j=1}^n a_{ij}\frac {\partial ^2 u}{\partial x_i^2 \partial x_j^2}$, it's transferable to $\nabla ^2 = \frac {\partial ^2}{...