Gradient estimates via linear and nonlinear potentials

By Frank Duzaar, Giuseppe Mingione. 2010

206 citations recorded by the source corpus.

We prove new potential and nonlinear potential pointwise gradient estimates for solutions to measure data problems, involving possibly degenerate quasilinear operators whose prototype is given by −Δpu=μ. In particular, no matter the nonlinearity of the equations considered, we show that in the case p⩽2 a pointwise gradient estimate is possible using standard, linear Riesz potentials. The proof is based on the identification of a natural quantity that on one hand respects the natural scaling of the problem, and on the other allows to encode the weaker coercivity properties of the operators considered, in the case p⩽2. In the case p>2 we prove a new gradient estimate employing nonlinear potentials of Wolff type.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7fd8b44c14f86b63de04b5bc3802b8c567a5e24d

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In Applied Mathematics

In Computational Theory and Mathematics

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Citations in the corpus

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.”
The Wolff gradient bound for degenerate parabolic equations (2014, background): “This result has been upgraded to the gradient level in [37] for the case p = 2 and then in [12, 13] for p ≥ 2− 1/n (see also [26, 27] for relevant developments), where the following estimate is proved: |Du(x0)| ≤ c B(x0,r) (|Du| + s) dx + cWμ1/p,p(x0, 2r), (1.”
Nonlinear Measure Data Problems (2011, methodology): “The first result of this section is the following comparison lemma, which extends the ones contained in [38, 77].”
Linear Potentials in Nonlinear Potential Theory (2013, background): “2) has been proved in [10], but in this case it does not improve (1.”
Potential estimates and gradient boundedness for nonlinear parabolic systems (2012, background): “Very recently, the possibility of extending such pointwise estimates to the gradient level has been observed starting from the paper [33] for nonlinear elliptic equations with linear growth (p = 2), and then extended to the general degenerate case in the papers [11], [14] and [15].”
Riesz Potentials and Nonlinear Parabolic Equations (2013, background): “T) ∩ L∞(ΩT) has a compact support. RIESZ POTENTIALS AND NONLINEAR PARABOLIC EQUATIONS 3 1.1. Intrinsic and explicit Riesz potential estimates. Very recently, in [27, 28, 38] for the case p ≥ 2 and in [13] for the subquadratic one, it has been shown that, surprisingly enough, the regularity theory of”
Riesz potential estimates for a general class of quasilinear equations (2015, methodology): “The technique permitting to obtain L estimates, being q “close” to one, for the difference Du − Dv via scaling arguments has appeared several time before this paper, see [14, 15, 34].”