Search Results

First quick comment. If you want to integrate on manifolds, you need some kind of volume form. You didn't say anything specific about your $dy$, but you did say that the operator $\mathbb{D}$ is self- …

A version of the Fredholm alternative should apply here. Basically, elements of the kernel of the adjoint operator induce conditions on $f$ for your equation to be solvable. Let $(v^k)$ be a basis for …

The following book might be helpful: Elliptic Partial Differential Equations: Volume 1: Fredholm Theory of Elliptic Problems in Unbounded Domains by Vitaly Volpert (Birkhäuser, 2011) http://boo …

Solutions of the Laplace equation $\Delta u = 0$ include harmonic polynomials, which grow at most polynomially, each $|u(x)| \le C |x|^N$ for some $N>0$ depending on $u$. Define $D[v] = e^{-g(x)} \Del …

If your equation can be written as $$ (\delta_{ij}\partial_t + B_{ij} \partial_x + c_{ij}) u_j = f_i , $$ where the $c_{ij}$ are constants and $f_i = f_i(t,x)$, then the most general solution can be …

Much has been written on higher order variational problems, but most of what I'm aware of concerns itself with geometric aspects like the generalization of the symplectic or Hamilton-Jacobi formalisms …

To my knowledge, the principal symbol of a non-linear differential operator is not discussed very often. When I have seen it discussed, the definition basically coincided with your approach 1. For exa …

In general, No. Your equation is overdetermined. In components, $\partial_j \Phi_i = B_{ji}$ (defining $B$ in terms of $A$), implying $\partial_k B_{ji} - \partial_j B_{ki} = 0$ (you need to interpret …

If I'm not miscalculating, the variational equation for $\varphi$ gives $\delta E_\epsilon/\delta\varphi = \partial^k (\rho^2 \partial_k \varphi) = 0$. Writing your integral identity as a flux through …

It's difficult for me to check the details at the moment, but the book Non-Homogeneous Boundary Value Problems and Applications by Lions and Magenes might also be helpful, perhaps more so for elliptic …

I think the answer is No. You are essentially asking the following: If $0$ is not an eigenvalue of the Dirac operator $D$ on a compact Riemannian manifold, then does the underlying Riemannian metric h …

I believe that the construction of the Green operator $G$ for $P$ is standard and involves two important steps. First, one has to construct a parametrix $Q$ for $P$ and then find a correction term tha …

I believe that the answer is Yes by way of the method of the parametrix. This equation is elliptic (the principal symbol is injective, being a power of the principal symbol of the Laplacian) so the pa …

Consider a determined linear system of differential order $k$ of the form $\sigma^{i_1\cdots i_k}_{ab}(x) \partial_{i_1} \cdots \partial_{i_k} u^b(x) + l.o.t = 0$. The coefficients $\sigma^{i_1\cdots …

The idea of the weights used in the ADN definition of the principal symbol is quite natural in the context of graded vector spaces or more generally graded modules. In the equation $u = Dv$, $u=(u_1,\ …