The final version of the mean value theorem for harmonic functions

1996 ◽  
Vol 59 (3) ◽  
pp. 247-252 ◽  
Author(s):  
V. V. Volchkov
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Final Version ◽  
The Mean

scholarly journals A converse to the mean value theorem for harmonic functions

1993 ◽  
Vol 171 (2) ◽  
pp. 139-163 ◽  
Author(s):  
Wolfhard Hansen ◽  
Nikolai Nadirashvili
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
The Mean

scholarly journals The strong maximum principle for the heat equation

1984 ◽  
Vol 27 (3) ◽  
pp. 297-299 ◽  
Author(s):  
David Colton
Keyword(s):  
Maximum Principle ◽  
Heat Equation ◽  
Parabolic Equations ◽  
Harmonic Functions ◽  
Mean Value ◽  
Strong Maximum Principle ◽  
Mean Value Theorem ◽  
Alternate Proof ◽  
The Maximum Principle ◽  
The Mean

The strong maximum principle for harmonic functions is usually arrived at by appealing to the mean value theorem (c.f. [2], p. 53). It is also of course possible simply to appeal to the Hopf maximum principle [2], but using sledge hammers to kill flies is generally viewed as aesthetically unpleasing. In contrast to the case of harmonic functions, the only proof of the strong maximum principle for the heat equation that is known to me is to invoke Nirenberg's strong maximum principle for parabolic equations [2]. As in the case of harmonic functions, it seems desirable to provide a direct proof of this result without having to go through the subtle comparison arguments that are employed in the more general case. The purpose of this note is to provide a proof of the strong maximum principle for the heat equation based on a mean value theorem for solutions of the heat equation which we derive below. Such an approach provides a straightforward and simple proof of the strong maximum principle which avoids most of the detailed estimates of the proof of the maximum principle for more general parabolic equations. Unfortunately the proof of the maximum principle for the heat equation using the mean value theorem is not as short as the proof in the corresponding case of harmonic functions. It nevertheless seems worthwhile to show that such an alternate proof is possible, and it is to this purpose that we address this paper.

A Lattice Summation Using the Mean Value Theorem for Harmonic Functions

SIAM Review ◽  
1984 ◽  
Vol 26 (1) ◽  
pp. 113-115 ◽  
Author(s):  
Paul K. Mazaika
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
The Mean

A converse of the mean value theorem for harmonic functions

1981 ◽  
Vol 36 (5) ◽  
pp. 159-160 ◽  
Author(s):  
A A Kosmodem'yanskii (Jr.)
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
The Mean

scholarly journals On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1303
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Faraidun Kadir Hamasalh
Keyword(s):  
Time Scale ◽  
Initial Value Problems ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Discrete Fractional Calculus ◽  
Initial Value ◽  
Forward Difference ◽  
The Mean ◽  
Monotonicity Analysis ◽  
Monotonicity Results

Monotonicity analysis of delta fractional sums and differences of order υ∈(0,1] on the time scale hZ are presented in this study. For this analysis, two models of discrete fractional calculus, Riemann–Liouville and Caputo, are considered. There is a relationship between the delta Riemann–Liouville fractional h-difference and delta Caputo fractional h-differences, which we find in this study. Therefore, after we solve one, we can apply the same method to the other one due to their correlation. We show that y(z) is υ-increasing on Ma+υh,h, where the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and then, we can show that y(z) is υ-increasing on Ma+υh,h, where the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) for each z∈Ma+h,h. Conversely, if y(a+υh) is greater or equal to zero and y(z) is increasing on Ma+υh,h, we show that the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) on Ma,h. Furthermore, we consider some related results for strictly increasing, decreasing, and strictly decreasing cases. Finally, the fractional forward difference initial value problems and their solutions are investigated to test the mean value theorem on the time scale hZ utilizing the monotonicity results.

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