/pdf/the-wiener-test-and-potential-estimates-for-quasilinear-3a1aejx6gh.pdf

The Wiener test and potential estimates for quasilinear elliptic equations

In this article, we study some new connections between Liouville-type theorems and local properties of nonnegative solutions to superlinear elliptic problems. Namely, we develop a general method for derivation of universal, pointwise, a priori estimates of local solutions from Liouville-type theorems, which provides a simpler and unified treatment for such questions. The method is based on rescaling arguments combined with a key “doubling” property, and it is different from the classical rescaling method of Gidas and Spruck [16]. As an important heuristic consequence of our approach, it turns out that universal boundedness theorems for local solutions and Liouville-type theorems are essentially equivalent. The method enables us to obtain new results on universal estimates of spatial singularities for elliptic equations and systems, under optimal growth assumptions that, unlike previous results, involve only the behavior of the nonlinearity at infinity. For semilinear systems of Lane-Emden type, these seem to be the first results on singularity estimates to cover the full subcritical range. In addition, we give an affirmative answer to the so-called Lane-Emden conjecture in three dimensions

Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems

We consider non-linear elliptic equations having a measure in the right hand side, of the type div a(x, Du) = μ, 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 Calderon-Zygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates. To the memory of Vic Mizel, mathematician and gentleman

/pdf/the-calderon-zygmund-theory-for-elliptic-problems-with-110z5h0yc5.pdf

The Calderón-Zygmund theory for elliptic problems with measure data

We consider non-linear elliptic equations having a measure in the right hand side, of the type $ \divo a(x,Du)=\mu, $ 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 Calder\'on-Zygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates.

The Calder\'on-Zygmund theory for elliptic problems with measure data

The strong maximum principle revisited

This paper is motivated by the observation that solutions to certain variational inequalities involving partial differential operators of the form divA(x, u, Vu) + B(x, u, Vu), where A and B are Borel measurable, are solutions to the equation divA(x, u, Vu) + B(x, u, Vu) = ,u for some nonnegative Radon measure ,i. Among other things, it is shown that if u is a Holder continuous solution to this equation, then the measure ,u satisfies the growth property j[B(x, r)] 1 is given by the structure of the differential operator. Conversely, if u is assumed to satisfy this growth condition, then it is shown that u satisfies a Harnack-type inequality, thus proving that u is locally bounded. Under -the additional assumption that A is strongly monotonic, it is shown that u is Holder continuous.

/pdf/local-behavior-of-solutions-of-quasilinear-elliptic-12m3befo9f.pdf

J.-M. Rakotoson

Papers

Local behavior of solutions of quasilinear elliptic equations with general structure