scholarly journals Remarks on Local Boundedness and Local Holder Continuity of Local Weak Solutions to Anisotropic p-Laplacian Type Equations

2016 ◽  
Vol 2 (1-2) ◽  
pp. 157-169 ◽  
Author(s):  
Emmanuele Dibenedetto ◽  
Ugo Gianazza ◽  
Vincenzo Vespri
Keyword(s):  
Weak Solutions ◽  
Hölder Continuity ◽  
Holder Continuity ◽  
Local Boundedness ◽  
Local Hölder Continuity

Local Hölder regularity for a doubly singular PDE

2018 ◽  
Vol 22 (03) ◽  
pp. 1850054
Author(s):  
Eurica Henriques
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Iteration Method ◽  
Hölder Continuity ◽  
Hölder Regularity ◽  
Holder Continuity ◽  
Singular Parabolic Equation ◽  
Local Hölder Continuity ◽  
Singular Pde ◽  
Bounded Weak Solutions

We establish the local Hölder continuity for the nonnegative bounded weak solutions of a certain doubly singular parabolic equation. The proof involves the method of intrinsic scaling and the parabolic version of De Giorgi’s iteration method.

scholarly journals Remarks on parabolic De Giorgi classes

2021 ◽  
Author(s):  
Naian Liao
Keyword(s):  
Harnack Inequality ◽  
Hölder Continuity ◽  
Holder Continuity ◽  
Type Estimate ◽  
Local Boundedness ◽  
Heavy Machinery ◽  
Weak Harnack Inequality ◽  
Logarithmic Type ◽  
Local Hölder Continuity

AbstractWe make several remarks concerning properties of functions in parabolic De Giorgi classes of order p. There are new perspectives including a novel mechanism of propagating positivity in measure, the reservation of membership under convex composition, and a logarithmic type estimate. Based on them, we are able to give new proofs of known properties. In particular, we prove local boundedness and local Hölder continuity of these functions via Moser’s ideas, thus avoiding De Giorgi’s heavy machinery. We also seize this opportunity to give a transparent proof of a weak Harnack inequality for nonnegative members of some super-class of De Giorgi, without any covering argument.

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