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

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


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

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 $\Omega$.


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