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Symmetrization Inequalities for Difference Equations on Graphs

✍ Scribed by Alexander R. Pruss

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
213 KB
Volume
22
Category
Article
ISSN
0196-8858

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✦ Synopsis

We prove symmetrization inequalities for positive solutions of not necessarily . linear difference equations of the form

where ⌬ is a discrete Laplacian, is a convex decreasing function, c is a positive function and is a real function, on subsets of X = Y, where X is a graph and Y Ε½ . is the line ‫,ήšβ€¬ the circle graph β€«ήšβ€¬ , the m-regular tree T , the line graph L T of m m m T , or the edge graph of the octahedron. Instead of ⌬, we may also have some m other operators, including the heat operator. The typical result says that we use a discrete version of Steiner symmetrization to symmetrize the domain on which the equation is defined, and if all the functions and boundary values involved are Ε½ . appropriately symmetrized as well, then the increasing convex means of u x, ΠΈ are increased for each fixed x g X.

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