2019
DOI: 10.48550/arxiv.1901.02673
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Stefano Buccheri

Abstract: We study existence and Lorentz regularity of distributional solutions to elliptic equations with either a convection or a drift first order term. The presence of such a term makes the problem not coercive. The main tools are pointwise estimates of the rearrangements of both the solution and its gradient.

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Cited by 2 publications
(2 citation statements)
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“…Such a technique was used by Weinberger in [Wei62] in order to show boundedness of solutions with vanishing trace to − div(A∇u) = − div f and − div(A∇u) = g, where f ∈ L p and g ∈ L p 2 , p > n. Another well known technique consists of a use of test functions that leads to bounds for the derivative of the integral of |∇u| 2 over superlevel sets of u, where u is a subsolution to Lu ≤ − div f + g. This bound, combined with Talenti's inequality [Tal76,estimate (40)], gives an estimate for the derivative of the decreasing rearrangement of u, leading to bounds for u in various spaces and comparison results. This technique has been used by many authors in order to study regularity properties of solutions to second order pdes, some works being [Buc19]. However, as we mentioned above, to the best of our knowledge, no local boundedness results have been deduced using this method so far.…”
Section: Past Workmentioning
confidence: 99%
“…Such a technique was used by Weinberger in [Wei62] in order to show boundedness of solutions with vanishing trace to − div(A∇u) = − div f and − div(A∇u) = g, where f ∈ L p and g ∈ L p 2 , p > n. Another well known technique consists of a use of test functions that leads to bounds for the derivative of the integral of |∇u| 2 over superlevel sets of u, where u is a subsolution to Lu ≤ − div f + g. This bound, combined with Talenti's inequality [Tal76,estimate (40)], gives an estimate for the derivative of the decreasing rearrangement of u, leading to bounds for u in various spaces and comparison results. This technique has been used by many authors in order to study regularity properties of solutions to second order pdes, some works being [Buc19]. However, as we mentioned above, to the best of our knowledge, no local boundedness results have been deduced using this method so far.…”
Section: Past Workmentioning
confidence: 99%
“…This idea is exhibited by the Pólya-Szegö inequality (see for example (1) in [BZ88]), and the fact that extremizers that achieve equality in the Sobolev inequality are radially symmetric functions [Tal76a]. Furthermore, this technique has been applied in many past works in order to show estimates of solutions to various problems concerning second order elliptic equations, for example in [Tal76b], [AT78], [AT81], [BM93], [DVP96], [DVP98], [ATLM99], [AFT00], and the more recent [Buc19].…”
Section: Introductionmentioning
confidence: 99%