Abstract
To study fine properties of certain smooth approximations \({u^\varepsilon}\) to a viscosity solution u of the infinity Laplacian partial differential equation (PDE), we introduce Green’s function \({\sigma^\varepsilon}\) for the linearization. We can then integrate by parts with respect to \({\sigma^\varepsilon}\) and derive various useful integral estimates. We are, in particular, able to use these estimates (i) to prove the everywhere differentiability of u and (ii) to rigorously justify interpreting the infinity Laplacian equation as a parabolic PDE.
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Communicated by C. Dafermos
Supported in part by NSF grants DMS-0500452 and DMS-1001724.
Supported in part by NSF grant DMS-1004595.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Evans, L.C., Smart, C.K. Adjoint Methods for the Infinity Laplacian Partial Differential Equation. Arch Rational Mech Anal 201, 87–113 (2011). https://doi.org/10.1007/s00205-011-0399-x
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DOI: https://doi.org/10.1007/s00205-011-0399-x