We give an imbedding theorem for the weak solutions of the Dirichlet problem (2) when is in certain Lorentz spaces: the main result (see Teorema 2) ensures the continuity of the weak solution when is in the Lorentz space ; from this fact, via a duality argument, we improve known results for the weak solutions of the equation (4).
We consider the homogeneous Dirichlet problem for nonlinear elliptic equations as where is a measure with bounded total variation. We fix structural conditions on functions , which ensure existence of solutions. Moreover, if is an function, we prove a uniqueness result under more restrictive hypotheses on the operator.
We give a pointwise bound for the rearrangement of a solution of the Dirichlet problem for second order elliptic equations with lower order terms by means of the solution of a "symmetrized" problem.
We give comparison results for solutions of variational inequalities, related to general elliptic second order operators, involving solutions of symmetrized problems, using Schwarz spherical symmetrization.
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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