Gradient estimates for a class of parabolic systems

By E. Acerbi, Giuseppe Mingione. 2007

451 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/08f281298795e763ce08f45c400ea889e4dfd925

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Elliptic equations with BMO nonlinearity in Reifenberg domains (2008, background): “The authors in [1] adopted an intrinsic geometric viewpoint, argued directly on certain Calderón–Zygmund type covering arguments, and so completely avoided the use of the maximal function.”
Optimal regularity for the Poisson equation (2009, methodology): “Recently Acerbi and Mingione [1] obtained local L, q ≥ p, gradient estimates for the degenerate parabolic p-Laplacian systems.”
Gradient estimates below the duality exponent (2010, background): “Specifically, starting from some arguments developed in [20],[43],[1] in different contexts, the level sets of the Hardy-Littlewood maximal function of |Du| are seen to be locally controlled by the level sets of a suitable Riesz potential operator, namely I1(|f |).”
Calderón–Zygmund theory for nonlinear elliptic problems with irregular obstacles (2012, methodology): “In this respect, it is suggested to follow the so-called maximal function-free technique which has been introduced in [1], in order to overcome such a lack of scaling.”
Nonlinear gradient estimates for elliptic equations of general type (2012, methodology): “This procedure makes a complete proof to be relatively technical and delicate (see, for instance, [2,4,8]).”
Lp solvability of divergence type parabolic and elliptic systems with partially BMO coefficients (2011, result): “In contrast to scalar equations, until quite recently, there are relatively few results of Lp theory for parabolic systems with discontinuous coefficients; see [18, 19, 10, 1, 8] and reference therein.”
Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations (2008, unknown): “For further discussion in this ﬁeld, see the work of Acerbi and Mingione [1], and the references therein.”
Potential estimates and gradient boundedness for nonlinear parabolic systems (2012, background): “2 is in a certain sense optimal, as in the case V = 0 it allows to recover the sharp interpolated L∞ bounds of DiBenedetto [5]; related bounds in L are available in [1].”
Nonlinear Measure Data Problems (2011, methodology): “This is typical when considering anisotropic problems, and similar deficit scaling exponents typically appear in the a priori estimates from [1, 2, 59].”
Global weighted estimates for the gradient of solutions to nonlinear elliptic equations (2013, methodology): “We would like to point out that the above higher integrability is a useful tool when treating nonlinear elliptic and parabolic problems, see for instance [1,2,16,22] and references therein.”
New perturbation methods for nonlinear parabolic problems (2012, methodology): “1) These will be called “intrinsic cylinders” as they will be usually employed in a context when the parameter λ is linked to the behavior of the solution of some equation on the same cylinder Qr according to the standard intrinsic geometry techniques (see for instance [9,1,17,15,16]).”
Gradient estimates via non-linear potentials (2009, background): “on-linear potentials - whose deﬁnition is built essentially using a standard family of balls and it is therefore “universal” - is not clear and will be the object of future investigation. We refer to [2] for global gradient estimates. In order to state our results we need some additional terminology”
Riesz Potentials and Nonlinear Parabolic Equations (2013, background): “16) are basic results of DiBenedetto [9] while we just remark that intrinsic geometries are nowadays a basic and common tool to treat degenerate parabolic equations [1, 17, 18].”