96
views
0
recommends
+1 Recommend
0 collections
    0
    shares
      • Record: found
      • Abstract: found
      • Article: found
      Is Open Access

      On the fundamental solution of an elliptic equation in nondivergence form

      Preprint

      Read this article at

      Bookmark
          There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

          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 \(x\to y\).

          Related collections

          Author and article information

          Journal
          0806.4108

          Analysis
          Analysis

          Comments

          Comment on this article