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An Upper Estimate for a Heat Kernel With Neumann Boundary Condition
Author(s) -
Lacey A. A.
Publication year - 1993
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/25.5.453
Subject(s) - mathematics , star (game theory) , combinatorics , heat kernel , neumann boundary condition , boundary (topology) , kernel (algebra) , upper and lower bounds , zero (linguistics) , function (biology) , measure (data warehouse) , mathematical analysis , linguistics , philosophy , database , evolutionary biology , computer science , biology
Using an upper solution we obtain a bound from above for the heat kernel ψ( x,y,t ) for a region Ω which is star‐shaped with respect to one of the points, say y . The estimate is for the Neumann problem and holds for short times. The form of the bound is ψ ( x , y , t ) ⩽( 4 π t )− N / 2 exp [ − |x − y | 2 / 4 t] + exp [ − d( x , y ) 2 / 4 t + O ( t − 1 / 2) ] ; moreover, for x ∈Ω\ Y ( y ), ψ ( x , y , t ) ⩽( 4 π t )− N / 2{ exp [ − |x − y | 2 / 4 t] + f ( x , y ) exp [ − d( x , y ) 2 / 4 t ] ( 1 + O ( t ) ) } . Here Y ( y ) is a closed subset of Ω ⋐ R N with measure zero, d(x,y) is the minimum distance between x and y via the boundary ∂Ω: d(x,y) = inf Z∈∂Ω (|x‐z| + |y‐z|), and f(.,y) is a positive function, continuous away from Y , and equal to unity on δΩ.