EUDML

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Regularity properties of solutions of elliptic equations in $R^{2}$ in limit cases

Angela Alberico; Vincenzo Ferone — 1995

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this paper the Dirichlet problem for a linear elliptic equation in an open, bounded subset of $R^{2}$ is studied. Regularity properties of the solutions are proved, when the data are $L^{1}$ -functions or Radon measures. In particular sharp assumptions which guarantee the continuity of solutions are given.

A symmetrization result for nonlinear elliptic equations.

Vincenzo Ferone; Basilio Messano — 2004

Revista Matemática Complutense

We consider a solution u of the homogeneous Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) = g(x,u) + f, where the principal term is a Leray-Lions operator defined on W (Ω). The function g(x,u) satisfies suitable growth assumptions, but no sign hypothesis on it is assumed. We prove that the rearrangement of u can be estimated by the solution of a problem whose data are radially symmetric.

A sharp isoperimetric inequality in the plane

Angelo Alvino; Vincenzo Ferone; Carlo Nitsch — 2011

Journal of the European Mathematical Society

We show that among all the convex bounded domain in $m a t h b b R^{2}$ having an assigned Fraenkel asymmetry index, there exists only one convex set (up to a similarity) which minimizes the isoperimetric deficit. We also show how to construct this set. The result can be read as a sharp improvement of the isoperimetric inequality for convex planar domain.