Skip to main content Accessibility help
×
Home

Estimates on the Green's function of second-order elliptic operators in ℝN

  • Adrian T. Hill (a1)

Abstract

Sharp upper and lower pointwise bounds are obtained for the Green's function of the equation

for λ> 0. Initially, in a Cartesian frame, it is assumed that . Pointwise estimates for the heat kernel of ut + Lu = 0, recently obtained under this assumption, are Laplace-transformed to yield corresponding elliptic results. In a second approach, the coordinate-free constraint is imposed. Within this class of operators, the equations defining the maximal and minimal Green's functions are found to be simple ODEs when written in polar coordinates, and these are soluble in terms of the singular Kummer confluent hypergeometric function. For both approaches, bounds on are derived as a consequence.

Copyright

References

Hide All
1Abramowitz, M. and Stegun, I. A.. Handbook of Mathematical Functions (Washington D.C.: National Bureau of Standards, 1964).
2Aronson, D. G.. Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 (1968), 607–94.
3Dautray, R. and Lions, J.-L.. Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3 (Berlin: Springer, 1990).
4Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.. Higher Transcendental Functions, Bateman Manuscript Project I (New York: McGraw-Hill, 1953).
5Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.. Tables of Integral Transforms, Bateman Manuscript Project I (New York: McGraw-Hill, 1954).
6Friedman, A.. Partial Differential Equations of Parabolic Type (Englewood Cliffs, NJ: Prentice-Hall, 1964).
7Henry, D.. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840 (Berlin: Springer, 1981).
8Hill, A. T.. Estimates on the heat kernel of parabolic equations with advection. SIAM J. Math. Anal. 28 (1997), 1309–16.
9Itô, S.. Diffusion Equations, Translations of Mathematical Monographs 114 (Providence, RI: American Mathematical Society, 1992).
10Pazy, A.. Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences Series 44 (New York: Springer, 1983).
11Pucci, C.. Operatori ellitici estremanti. Ann. Mat. Pura Appl. (4) 72 (1966), 141–70.
12Shimakura, N.. Partial Differential Operators of Elliptic Type, Translations of Mathematical Monographs 99 (Providence, RI: American Mathematical Society, 1992).

Estimates on the Green's function of second-order elliptic operators in ℝN

  • Adrian T. Hill (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed