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Expansion of the fundamental solution of a second-order elliptic operator with analytic coefficients

Federico Franceschini, Federico Glaudo

Abstract

Let $L$ be a second-order elliptic operator with analytic coefficients defined in $B_1\subseteq\mathbb R^n$. We construct explicitly and canonically a fundamental solution for the operator, i.e., a function $u:B_{r_0}\to\mathbb R$ such that $Lu=δ_0$. As a consequence of our construction, we obtain an expansion of the fundamental solution in homogeneous terms (homogeneous polynomials divided by a power of $|x|$, plus homogeneous polynomials multiplied by $\log(|x|)$ if the dimension $n$ is even) which improves the classical result of F. John (1950). The control we have on the "complexity" of each homogeneous term is optimal and in particular, when $L$ is the Laplace-Beltrami operator of an analytic Riemannian manifold, we recover the construction of the fundamental solution due to K. Kodaira (1949). The main ingredients of the proof are a harmonic decomposition for singular functions and the reduction of the convergence of our construction to a nontrivial estimate on weighted paths on a graph with vertices indexed by $\mathbb Z^2$.

Expansion of the fundamental solution of a second-order elliptic operator with analytic coefficients

Abstract

Let be a second-order elliptic operator with analytic coefficients defined in . We construct explicitly and canonically a fundamental solution for the operator, i.e., a function such that . As a consequence of our construction, we obtain an expansion of the fundamental solution in homogeneous terms (homogeneous polynomials divided by a power of , plus homogeneous polynomials multiplied by if the dimension is even) which improves the classical result of F. John (1950). The control we have on the "complexity" of each homogeneous term is optimal and in particular, when is the Laplace-Beltrami operator of an analytic Riemannian manifold, we recover the construction of the fundamental solution due to K. Kodaira (1949). The main ingredients of the proof are a harmonic decomposition for singular functions and the reduction of the convergence of our construction to a nontrivial estimate on weighted paths on a graph with vertices indexed by .

Paper Structure

This paper contains 7 sections, 19 theorems, 124 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. On the covariance of the construction
  4. Acknowledgements
  5. Harmonic decomposition
  6. Combinatorial bounds for weighted paths in Z
  7. Proof of the main theorems

Key Result

Theorem 1.1

For $n\geq 2$, consider the elliptic differential operator where $A_{ij},b_i,c$ are analytic in the unit ball $B_1$ and $A(x)$ is a positive symmetric matrix for every $x\in B_1$. Then there is a radius $r_0=r_0(n, A, b, c)>0$ such that the following statement hold. There is an analytic function $v\colon B_{r_0}\to\mathbb R$ (which is null if the dimension converges absolutely in the $C^{\infty}

Figures (1)

  • Figure 1: Diagrams representing $G_1$ and $G_2$.

Theorems & Definitions (52)

  • Theorem 1.1: John1950
  • Theorem 1.2: Kodaira49
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • ...and 42 more