Abstract
A-priori pointwise estimates to difference-quotients of solutions to elliptic or parabolic equations can be obtained by using the maximum property of appropriate higher-dimensional operators. This method, introduced by Brandt, is here used for a simple derivation of the interior Schauder estimates for second-order parabolic differential equations. The same derivation is applicable also for the analogous finite-difference equations.
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References
A. Brandt,Interior estimates for second-order elliptic differential (or finite-difference) equations via the maximum principle, Israel J. Math.7 (1969) 95–121.
A. Brandt,Generalized maximum principles for second order difference equations, to appear.
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A. Friedman,Partial differential equations of parabolic type, Prentice-Hall, 1964.
V. Thomée,Discrete interior Schauder estimates for elliptic difference operators, SIAM J. Number. Anal.,5 (1968), 626–645.
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Brandt, A. Interior schauder estimates for parabolic differential- (or difference-) equations via the maximum principle. Israel J. Math. 7, 254–262 (1969). https://doi.org/10.1007/BF02787619
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DOI: https://doi.org/10.1007/BF02787619