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Seminario / Romani

Martedì 14 dicembre 2021, alle ore 15.00 in aula C
Giulio ROMANI (Università degli Studi di Milano-Bicocca)
terrà il seguente seminario

"Failure of local positivity for general higher-order elliptic operators"
The positivity preserving property (Lu = f ≥ 0 --> u ≥ 0 ?) for second-order elliptic equations is a well-known consequence of the maximum principle. In the higher-order setting such behaviour is often spoiled by the influence of the boundary conditions, and in general the answer is negative. However, one still expects that, applying an extremely concentrated right-hand side − a δ-distribution −, then close to this point the solution should respond in the same direction. In large dimensions n ≥ 2m such local question can be rewritten in terms of the positivity of a singular fundamental solution near to its pole. Hence, the local positivity is ensured for the polyharmonic operator (−∆)^m and, more in general, for powers of second-order operators.
In this talk, we show that such behaviour cannot be in general expected for any elliptic higher-order operator, even in the case of constant coefficients. Indeed, by means of explicit closed expressions for the fundamental solutions in terms of its symbol, together with an inductive argument by space dimension, we prove that positivity near the unbounded singularity persists only in the special dimensions n = 2m and n = 2m + 1, while for any m ≥ 2 and n ≥ 2m + 2 there exist operators of order 2m whose fundamental solutions are sign-changing near the pole.
The results in this talk are based on a joint work with H.-C. Grunau and G. Sweers.

Tutti gli interessati sono invitati a partecipare.
Per ulteriori informazioni rivolgersi a:  Cristina.Tarsi@unimi.it
09 dicembre 2021
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