##
**Elliptic second order equations.**
*(English)*
Zbl 0726.35036

From the author’s introduction: “These lectures deal primarily with elliptic second-order equations in non-divergence form; for the sake of simplicity we consider through most of the lectures linear equations with leading terms only, namely the equation (1) \(a_{ij}(x)D_{ij}u=f.\)

We emphasize that no smoothness assumptions are made on the coefficients. Our main concern here is to describe some qualitative properties of the solutions of (1), such as Harnack inequality, Hölder regularity theory and Calderon-Zygmund \(L^ p\) estimates.

(...). We shall prove here the same type of results, but asking the coefficients and the right-hand side to be under control in \({\mathbb{R}}^ n\). Analogous results will be proved in Sobolev classes. But it is known that neither Hölder nor Sobolev classes are adequate to describe all the solutions of elliptic second order equations with non-smooth coefficients. Then we shall consider solutions unendowed with derivatives, that satisfy the equation in a suitable generalized sense (viscosity solutions).”

We emphasize that no smoothness assumptions are made on the coefficients. Our main concern here is to describe some qualitative properties of the solutions of (1), such as Harnack inequality, Hölder regularity theory and Calderon-Zygmund \(L^ p\) estimates.

(...). We shall prove here the same type of results, but asking the coefficients and the right-hand side to be under control in \({\mathbb{R}}^ n\). Analogous results will be proved in Sobolev classes. But it is known that neither Hölder nor Sobolev classes are adequate to describe all the solutions of elliptic second order equations with non-smooth coefficients. Then we shall consider solutions unendowed with derivatives, that satisfy the equation in a suitable generalized sense (viscosity solutions).”

Reviewer: M.Chicco (Genova)

### MSC:

35J15 | Second-order elliptic equations |

35R05 | PDEs with low regular coefficients and/or low regular data |

35B45 | A priori estimates in context of PDEs |

35B65 | Smoothness and regularity of solutions to PDEs |

### Keywords:

non-divergence form; Harnack inequality; Hölder regularity; Calderon- Zygmund \(L^ p\) estimates; viscosity solutions
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\textit{L. Caffarelli}, Rend. Semin. Mat. Fis. Milano 58, 253--284 (1988; Zbl 0726.35036)

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### References:

[1] | [C-L]Crandall M. G. andLions P. L., “Condition d’unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre{”, Comptes-Rendus Paris 292 (1981) pp. 183–186.} |

[2] | [G-T]Gilbarg D. andTrudinger N. S.,Elliptic Partial Differential Equations of Second Order, 2nd Ed., Springer, New York, 1983. |

[3] | [K-S]Krypov N. V. andSafonov M. V., “An estimate on the probability that a diffusion hits a set of positive measure{”, Soviet Math. 20 (1979) pp. 253–256.} |

[4] | [P]Pucci C., “Limitazioni per soluzioni di equazioni ellittiche{”, Ann. Mat. Pura Appl. 74 (1966) pp. 15–30.} · Zbl 0144.35801 |

[5] | [J]Jensen R., “The Maximum Principle for Viscosity Solutions of Fully Nonlinear Second Order Partial Differential Equations{” (to appear).} · Zbl 0708.35019 |

[6] | [I]Ishi H., “On uniqueness and viscosity solutions of fully nonlinear second order elliptic P.D.E.’s{”, preprint.} |

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