On the fundamental solution of an elliptic equation in nondivergence form

TL;DR

This paper studies the fundamental solution of elliptic operators in nondivergence form with continuous coefficients, analyzing its existence, asymptotics, and the role of an integral function I(x,y) in determining solution behavior.

Contribution

It constructs explicit fundamental solutions under square Dini continuity conditions and characterizes their asymptotic behavior based on the limit of the integral I(x,y).

Findings

If I(x,y) approaches a finite limit, the fundamental solution resembles that of a constant coefficient operator.

If I(x,y) tends to -infinity, the solution violates the maximum principle and is unbounded near the singularity.

The paper provides conditions under which the fundamental solution exists and describes its asymptotic properties.

Abstract

We consider the existence and asymptotics for the fundamental solution of an elliptic operator in nondivergence form, ${\mathcal L}(x,\del_x)=a_{ij}(x)\del_i\del_i$, for $n\geq 3$. We assume that the coefficients have modulus of continuity satisfying the square Dini condition. For fixed $y$, we construct a solution of ${\mathcal L}Z_y(x)=0$ for $0<|x-y|<\e$ with explicit leading order term which is $O(|x-y|^{2-n}e^{I(x,y)})$ as $x\to y$, where $I(x,y)$ is given by an integral and plays an important role for the fundamental solution: if $I(x,y)$ approaches a finite limit as $x\to y$, then we can solve ${\mathcal L}(x,\del_x)F(x,y)=\de(x-y)$, and $F(x,y)$ is asymptotic as $x\to y$ to the fundamental solution for the constant coefficient operator ${\mathcal L}(y,\del_x)$. On the other hand, if $I(x,y)\to -\infty$ as $x\to y$ then the solution $Z_y(x)$ violates the "extended maximum principle" of Gilbarg & Serrin \cite{GS} and is a distributional solution of ${\mathcal L}(x,\del_x)Z_y(x)=0$ for $|x-y|<\e$ although $Z_y$ is not even bounded as $x\to y$.