Elliptic operators with unbounded diffusion coefficients in $L^p$ spaces

Metafune, Giorgio¹  ; Spina, Chiara¹

¹ Dipartimento di Matematica “Ennio De Giorgi” Università del Salento C.P. 193 73100, Lecce, Italia

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 2, pp. 303-340

Résumé

We prove that, for $N\ge 3$, $\alpha >2$, $\frac{N}{N-2}<p<\infty$, the operator $Lu=m\left(x\right)(1+|x{|}^{\alpha })\Delta u$ generates an analytic semigroup in $L^p$ which is contractive if and only if $p\ge \frac{N+\alpha -2}{N-2}$. Moreover, for $\alpha <\frac{N}{p^{\text{'}}}$, we provide an explicit description of the domain. Spectral properties of the operator $L$ are also obtained.

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Publié le : 2018-06-21

Classification : 47D07, 35B50, 35J25, 35J70