In this paper the Dirichlet problem for a linear elliptic equation in an open, bounded subset of is studied. Regularity properties of the solutions are proved, when the data are -functions or Radon measures. In particular sharp assumptions which guarantee the continuity of solutions are given.
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.
We show that among all the convex bounded domain in 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.
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