We consider non-linear elliptic equations having a measure in the right-hand side, of the type and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable Calderón-Zygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates.
Pointwise gradient bounds via Riesz potentials like those available for the Poisson equation actually hold for general quasilinear equations.
La teoria di Calderón-Zygmund per equazioni ellittiche e paraboliche lineari ammette un analogo non lineare che si è andato man mano delineando sempre più chiaramente negli ultimi anni. Di seguito si discutono alcuni risultati validi in questo ambito.
I am presenting a survey of regularity results for both minima of variational integrals, and solutions to non-linear elliptic, and sometimes parabolic, systems of partial differential equations. I will try to take the reader to the Dark Side...
We describe a few results obtained in [10], concerning the possibility of estimating solutions of quasilinear elliptic equations via nonlinear potentials.
We consider the integral functional under non standard growth assumptions of -type: namely, we assume that , a relevant model case being the functional . Under sharp assumptions on the continuous function we prove regularity of minimizers both in the scalar and in the vectorial case, in which we allow for quasiconvex energy densities. Energies exhibiting this growth appear in several models from mathematical physics.
We prove partial regularity for minimizers of the functional where the integrand f(x,u,ξ) is quasiconvex with subquadratic growth: , p < 2. We also obtain the same results for ω-minimizers.
The spatial gradient of solutions to non-homogeneous and degenerate parabolic equations of -Laplacean type can be pointwise estimated by natural Wolff potentials of the right hand side measure.
We describe some recent results obtained in [29], where we prove regularity theorems for sub-elliptic equations in (horizontal) divergence form defined in the Heisenberg group, and exhibiting polynomial growth of order p. The main result tells that when solutions to possibly degenerate equations are locally Lipschitz continuous with respect to the intrinsic distance. In particular, such result applies to p-harmonic functions in the Heisenberg group. Explicit estimates are obtained, and eventually...
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