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1 vote
0 answers
76 views

PDE on an open ball with prescribed value on some open subsets

Suppose that we have a $r$-order differential operator $L:C^r(B^n, \mathbb{R})\to C^0(B^n,\mathbb{R})$ where $B^n $ is the open unitary ball in $\mathbb{R}^n$ (we can assume for simplicity that it ...
Overflowian's user avatar
  • 2,533
5 votes
0 answers
143 views

Extension of elliptic complex to an exact sequence

This questions concerns elliptic complexes and is closely related to Green's operator of elliptic differential operator. Let $T_f:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial ...
Tobias Diez's user avatar
  • 5,824
2 votes
1 answer
127 views

Positive form for a homogeneous elliptic pde

I have a pde of the following form: \begin{align} &P(x,D)u = f \text{ on } \Omega, \\ &P(x,D) = \sum\limits_{|\alpha|=2m}a_\alpha(x)D^{\alpha}, \end{align} where one can assume that $f$ ...
Fedor Goncharov's user avatar
1 vote
0 answers
158 views

Domain of operator

Let be $\lambda\in C^{*}$. Consider the following operator: $ T_{\lambda}=-\Delta_{R^{2}}++\frac{\lambda^{2} }{4} (x^{2}+y^{2})+i\lambda N$, where $N=(x \frac{d }{dy} -y \frac{d }{dx})$ , ...
Fadil Kikawi's user avatar
17 votes
4 answers
3k views

Green's operator of elliptic differential operator

Let $P:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial differential operator, with index $=0$ and closed image of codimension $=1$, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections ...
mdg's user avatar
  • 376