Gradient estimates via non-linear potentials

By Frank Duzaar, Giuseppe Mingione. 2009

302 citations recorded by the source corpus.

We present pointwise gradient bounds for solutions to $p$-Laplacean type non-homogeneous equations employing non-linear Wolff type potentials, and then prove similar bounds, via suitable caloric potentials, for solutions to parabolic equations.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/1d4c1c3eba206f3ba061a8cb7b0b45278ce00718

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In Applied Mathematics

1. Giuseppe Mingione
2. Tuomo Kuusi
3. Andrea Cianchi
4. Vladimir G. Maz'ya
5. Frank Duzaar
6. Cristiana De Filippis
7. Paolo Baroni
8. Iwona Chlebicka

In Computational Theory and Mathematics

1. Giuseppe Mingione
2. Tuomo Kuusi
3. Andrea Cianchi
4. Frank Duzaar
5. Vladimir G. Maz'ya
6. Iwona Chlebicka
7. Cristiana De Filippis
8. Verena Bogelein

Citations in the corpus

• Riesz Potentials and Nonlinear Parabolic Equations (2013, background): “never Qr(x0,t0) ⊂ ΩT is a standard parabolic cylinder such that r ≤ R0; here R0 is the radius introduced in Theorem 1.1. For the case p = 2 the previous estimate recovers the main parabolic result in [12]. An immediate consequence of Theorem 1.2 is the following global bound via classical, non-trunc”
• Pointwise Calderón–Zygmund gradient estimates for the p -Laplace system (2015, unknown): “The study of pointwise elliptic gradient regularity has received an impulse from the papers [49] and [27], where Havin-Maz’ya-Wolff nonlinear potentials have been shown to yield precise estimates for the gradient of local solutions to nonlinear p-Laplacian type equations, but with right-hand side in”
• Local Lipschitz regularity for degenerate elliptic systems (2009, background): “e is the so called Wolff potential defined by WV β,p(x,R) := ZR 0 |V|(B(x,)) n−βp 1 p−1 d β<n/p, where V is a Radon measure. In this respect pointwise estimates for solutions are obtained in [22, 35, 30, 11]. The peculiarity of the non-linear potential introduced in (1.11) relies in that it allow”
• The Wolff gradient bound for degenerate parabolic equations (2014, methodology): “Ultimately, the iteration methods introduced in [17, 40, 41, 36, 37, 12] cannot be any longer applied.”
• Morrey global bounds and quasilinear Riccati type equations below the natural exponent (2014, unknown): “, in [24, 54].”
• Linear Potentials in Nonlinear Potential Theory (2013, methodology): “1 somehow unexpected is that, starting from the seminal papers of Kilpeläinen & Malý [18,19], with different approaches offered by Trudinger & Wang [38,39] (that also worked in the subelliptic setting, starting by the ideas in [37,36]) and Korte & Kuusi [20] and Duzaar & Mingione [9], it is a standa”
• Gradient continuity estimates (2010, methodology): “In [ 15 ] we proved the following gradient pointwise estimate for solutions to (1):”