# Generalization of the Pucci extremal operators.?

Let $$S$$ the set of symmetric matrices, and $$S_{\lambda,\Lambda}$$ the set of symmetric matrices whose eingenvalues belong to $$[\lambda,\Lambda]$$, we define the Pucci's extremal operators as

$$\mathcal{M}^{-}(X,\lambda,\Lambda)= \inf\limits_{A \in S_{\lambda,\Lambda}} tr(AX)$$

and

$$\mathcal{M}^{+}(X,\lambda,\Lambda)= \inf\limits_{A \in S_{\lambda,\Lambda}} tr(AX)$$

What would happen if the symmetric matrix had eigenvalues ​​depending continuously on $$x$$? Was there a generalization for Pucci operators? In other words, if there are $$\lambda_{x}$$ and $$\Lambda_{x}$$ eigenvalues of a symmetric continuous matrix m $$A_{x} \in M_{n}$$, where $$x \in \Omega \subset \mathbb{R}^{n}$$ limited domain such that $$\lambda_{x} \leq \Lambda_{x} \forall x \in \Omega$$ , can we still talk about Pucci operators?

My first thought woluld be the following: intuitively, Pucci's extremal operators are "worst case scenarios", in the sense that $$\mathcal{M}^-$$ gives you the most upward bending and $$\mathcal{M}^+$$ gives you the most downward bending. This is easiest seen by using the following (equivalent form): $$\mathcal{M}^-(X)=\lambda\sum_{e_i>0}e_i +\Lambda\sum_{e_i<0}e_i\quad \mathcal{M}^+(X)=\Lambda\sum_{e_i>0}e_i +\lambda\sum_{e_i<0}e_i$$ where $$\{e_i\}_i$$ are the eigenvalues of $$X$$. If the ellipticity constants would depend on $$x$$ you would wind up with something like $$\mathcal{M}^-(x,X)=\lambda(x)\sum_{e_i>0}e_i +\Lambda(x)\sum_{e_i<0}e_i\quad \mathcal{M}^+(x,X)=\Lambda(x)\sum_{e_i>0}e_i +\lambda(x)\sum_{e_i<0}e_i,$$ right?
Now, at the heart of ellipticity lies the fact that $$\lambda$$ (and hence also $$\Lambda$$) are strictly positive. Then, my guess is that the actual difficulty (and probably source of special interest) for developing a theory there would be the possibility of $$\lambda(x)$$ being allowed to get arbitrarily close to $$0$$ (degenerate) or $$\Lambda(x)$$ being unbounded (singular) in $$\Omega$$.