zbMATH — the first resource for mathematics

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)

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