A mean value theorem for the heat equation

@inproceedings{Fulks1966AMV,
  title={A mean value theorem for the heat equation},
  author={Watson Fulks},
  year={1966}
}
The Gauss mean value theorem and its converse, due to Koebe, characterize solutions of Laplace's equation [2]. In view of the strong analogy between Laplace's equation and the heat equation it seems reasonable to expect an analogue of the Gauss-Koebe result to hold for the heat equation. The purpose of this paper is to present such a result. For simplicity we work in two dimensions, though it is clear from the calculations that the result is independent of dimensions. Murakami [3] has used a… 
View via Publisher
ams.org
40 Citations
Highly Influential Citations
2
Background Citations
10
Methods Citations
2
Results Citations
1

40 Citations

A Pizzetti-type formula for the heat operator
Abstract.We provide an asymptotic expansion of the integral mean of a smooth function over the Heat ball. Namely we generalize to the Heat operator the so-called Pizzetti’s Formula, which expresses…
On the Potential Theory of the KOLMOGOROV Equation
We prove a mean value theorem for the solutions of the backward KOLMOGOROV equation. The mean values are taken over the level surfaces of the fundamental solution of the formally adjoint operator…
Bruno Pini and the Parabolic Harnack Inequality: The Dawning of Parabolic Potential Theory
In this article, we describe the pioneering work of Bruno Pini toward the modern Potential Analysis of second order parabolic Partial Differential Equations. We mainly focus on the parabolic Harnack…
Introduction to Heat Potential Theory
This book is the first to be devoted entirely to the potential theory of the heat equation, and thus deals with time dependent potential theory. Its purpose is to give a logical, mathematically…
Initial-boundary value problem for the heat equation—A stochastic algorithm
The initial-boundary value problem for the heat equation is solved by using an algorithm based on a random walk on heat balls. Even if it represents a sophisticated generalization of the Walk on…
02 2 INVERSE MEAN VALUE PROPERTIES ( A SURVEY )
Inverse mean value properties of harmonic functions (C2-solutions of the Laplace equation; see [1], p. 25, about the origin of the term ‘harmonic’) are well known for balls and spheres; see the…
Volume mean densities for the heat equation
It is shown that, for solid caps D of heat balls in ℝd + 1 with center z0 = (0, 0), there exist Borel measurable functions w on D such that inf w(D) > 0 and ∫v(z)w(z) dz ≤ v(z0), for every…
Regular Boundary Points and Exit Distributions for Parabolic Differential Operators
We show that the regularity of a boundary point for a parabolic differential operator in divergence form is under some geometric assumptions equivalent to the property that the density of the exit…
Mean value theorems for solutions of linear partial differential equations
We consider generalized mean value theorems for solutions of linear differential equations with constant coefficients and zero right-hand side which satisfy the following homogeneity condition with…
MEAN VALUE THEOREMS ON MANIFOLDS
We derive several mean value formulae on manifolds, generalizing the classical one for harmonic functions on Euclidean spaces as well as later results of Schoen-Yau, Michael-Simon, etc, on curved…
...
...

References

SHOWING 1-3 OF 3 REFERENCES
A strong maximum principle for parabolic equations
Semi-linear partial differential equations of parabolic type, Funkcial
  • Ekvac
  • 1960
Foundations of potential theory