Caccioppoli estimates and very weak solutions of elliptic equations

Iwaniec, Tadeusz ; Sbordone, Carlo

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 14 (2003) no. 3, pp. 189-205.

Résumé

Caccioppoli estimates are instrumental in virtually all analytic aspects of the theory of partial differential equations, linear and nonlinear. And there is always something new to add to these estimates. We emphasize the fundamental role of the natural domain of definition of a given differential operator and the associated weak solutions. However, we depart from this usual setting (energy estimates) and move into the realm of the so-called very weak solutions where important new applications lie. We carry out this task deliberately with a restricted generality in interest of readability, and we hope it pays off handsomely in mathematical insights.

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