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On the Aleksandrov–Bakel’man–Pucci Estimate for Some Elliptic and Parabolic Nonlinear Operators

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Abstract

In this work we prove the Aleksandrov–Bakel’man–Pucci estimate for (possibly degenerate) nonlinear elliptic and parabolic equations of the form

$$ -{\rm div} \left( F\left( \nabla u(x)\right) \right) =f\left(x\right) \quad {\rm in}\,\Omega \subset \mathbb{R}^{n} $$

and

$$ u_{t}(x,t)-{\rm div} \left( F\left( \nabla u(x,t)\right) \right) =f\left( x,t\right) \quad {\rm in}\,Q\subset \mathbb{R}^{n+1} $$

for F a \({\fancyscript{C}^1}\) monotone field under some suitable conditions. Examples of applications such as the p-Laplacian and the Mean Curvature Flow are considered, as well as extensions of the general results to equations that are not in divergence form, such as the m-curvature flow.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Roberto Argiolas & Fernando Charro

  2. Departamento de Matemáticas, Universidad Autónoma de Madrid and ICMAT, 28049, Madrid, Spain

    Ireneo Peral

Authors
  1. Roberto Argiolas
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  2. Fernando Charro
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  3. Ireneo Peral
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Correspondence to Ireneo Peral.

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Communicated by F. Lin

Dedicated to Sandro Salsa on his 60th birthday, with our friendship.

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Argiolas, R., Charro, F. & Peral, I. On the Aleksandrov–Bakel’man–Pucci Estimate for Some Elliptic and Parabolic Nonlinear Operators. Arch Rational Mech Anal 202, 875–917 (2011). https://doi.org/10.1007/s00205-011-0434-y

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