The fundamental solution of the heat equation associated with the $$\bar \partial $$ -neumann problem-neumann problem

1978 ◽  
Vol 34 (1) ◽  
pp. 265-274 ◽  
Author(s):  
Nancy K. Stanton
Keyword(s):  
Fundamental Solution ◽  
Heat Equation ◽  
Neumann Problem

The Explicit Solution of the -Neumann Problem in a Non-Isotropic Siegel Domain

1997 ◽  
Vol 49 (6) ◽  
pp. 1299-1322 ◽  
Author(s):  
Jingzhi Tie
Keyword(s):  
Differential Equation ◽  
Fundamental Solution ◽  
Heat Equation ◽  
Heisenberg Group ◽  
Neumann Problem ◽  
Explicit Solution ◽  
Laplacian Operator ◽  
Siegel Domain ◽  
The Neumann Problem ◽  
Image Position

AbstractIn this paper, we solve the-Neumann problem on (0, q) forms, 0 ≤ q ≤ n, in the strictly pseudoconvex non-isotropic Siegel domain:where aj> 0 for j = 1,2, . . . , n. The metric we use is invariant under the action of the Heisenberg group on the domain. The fundamental solution of the related differential equation is derived via the Laguerre calculus. We obtain an explicit formula for the kernel of the Neumann operator. We also construct the solution of the corresponding heat equation and the fundamental solution of the Laplacian operator on the Heisenberg group.

scholarly journals Kesten's bound for subexponential densities on the real line and its multi-dimensional analogues

2018 ◽  
Vol 50 (2) ◽  
pp. 373-395 ◽  
Author(s):  
Dmitri Finkelshtein ◽  
Pasha Tkachov
Keyword(s):  
Fundamental Solution ◽  
Heat Equation ◽  
Probability Density ◽  
Real Line ◽  
The Real ◽  
Probability Densities ◽  
Tail Asymptotic

Abstract We study the tail asymptotic of subexponential probability densities on the real line. Namely, we show that the n-fold convolution of a subexponential probability density on the real line is asymptotically equivalent to this density multiplied by n. We prove Kesten's bound, which gives a uniform in n estimate of the n-fold convolution by the tail of the density. We also introduce a class of regular subexponential functions and use it to find an analogue of Kesten's bound for functions on ℝd. The results are applied to the study of the fundamental solution to a nonlocal heat equation.

FUNDAMENTAL SOLUTION OF HEAT EQUATION IN CONDENSED PHASE

2018 ◽  
Vol 100 (2) ◽  
pp. 167-180
Author(s):  
E. V. Alves ◽  
M. J. Alves
Keyword(s):  
Fundamental Solution ◽  
Heat Equation ◽  
Condensed Phase

Asymptotic behaviour of the fundamental solution to ∂ u /∂ t = — ( — ∆) m u

1993 ◽  
Vol 441 (1912) ◽  
pp. 423-432 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Fundamental Solution ◽  
Heat Equation ◽  
Asymptotic Behaviour ◽  
Infinite Number ◽  
Fourier Integral ◽  
Number Of Zeros

An asymptotic expansion is derived for the Fourier integral f ^ ( x ) = 1 ( 2 π ) n / 2 ∫ R n exp ( − | q | 2 m + i x ⋅ q ) d q , x ε R n as | x | →∞, where m is a positive integer. From this, it is deduced that the fundamental solution to the ‘heat’ equation ∂ u / ∂ t = − ( − Δ ) m u has an infinite number of zeros tending to infinity.

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