The Calderón-Zygmund theory for elliptic problems with measure data

By Giuseppe Mingione. 2006

368 citations recorded by the source corpus.

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

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/53ccc6d6c6ef0772da2753c3205b86d377fb6f0d

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Citations in the corpus

Fractional p-eigenvalues (2013, neutral): “The interested reader is referred to [16, 17, 8].”
Nonlinear Muckenhoupt-Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations (2013, background): “The following important comparison lemma involving an estimate “below the natural growth exponent” was established in [26] in the case p ≥ 2 and in [10] in the case 2− 1/n < p < 2.”
Hitchhiker's guide to the fractional Sobolev spaces (2011, background): “equations, are now experiencing impressive applications in different subjects, such as, among others, the thin obstacle problem [87, 69], optimization [37], finance [26], phase transitions [2, 14, 88, 40, 45], stratified materials [83, 23, 24], anomalous diffusion [68, 98, 65], crystal dislocation [”
Local behavior of fractional $p$-minimizers (2015, positive): “Also, it is worth mentioning that other fractional Caccioppoli-type inequalities have been recently used in different contexts (see, for instance, [21, 22, 9]), although none of them takes into account the tails.”
Linear Potentials in Nonlinear Potential Theory (2013, background): “We refer for instance to [30] for the last inequality and more properties of the map V (·).”
Sharp regularity for general Poisson equations with borderline sources (2011, background): “As usual in the literature, we could also include a parameter s 0 as to distinguish the model p-Laplacian operator (s = 0) from the nondegenerate one (s > 0), see for instance [14,15].”
Higher integrability for parabolic systems with non-standard growth and degenerate diffusions (2011, methodology): “This idea was first successfully applied in the context of measure data problems in [36].”
Gradient regularity for elliptic equations in the Heisenberg group (2007, neutral): “3), see also [42], Section 2.”