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Classical, viscosity and average solutions for PDE’s with nonnegative characteristic form. (English) Zbl 1098.35052
Let \(L\) be the following linear differential second order operator \[ \sum_{i,j=1}^{n}a_{ij}(x)\partial _{x_{i}x_{j}}+\sum_{i=1}^{n}b_{i}(x)\partial _{x_{i}},\quad x\in \Omega , \] where \(\Omega \) is an open subset of \(\mathbb{R}^{n},\) the matrix \((a_{ij}(x))_{1\leq i,j\leq n}\) is symmetric and non-negative definite at any \(x\in \Omega ,\) and the functions \(a_{ij}\) and \(b_{i}\) are continuous.
The authors introduce a notion of asymptotic-average solution for \(L,\) called \(AL\)-solution, under the assumption that the operator \(L\) satisfies an asymptotic representation formula. This formula is suggested by the classical Pizzeti formula. This \(AL\)-solution is compared with other notions of weak solution. It is proved that, if the operator \(L\) satisfies suitable conditions, the notions of classical, viscosity and \(AL\)-solution are equivalent. Some examples are given illustrating the results of the paper.
Reviewer: C. Bouzar (Oran)

35D05 Existence of generalized solutions of PDE (MSC2000)
35J70 Degenerate elliptic equations