On the fundamental solution of an elliptic equation in nondivergence form
Vladimir Gilelevich Maz'ya, Robert C. McOwen · Translations - American Mathematical Society/Translations · 2010
Abstract. We consider the existence and asymptotics for the fundamental solution of an elliptic operator in nondivergence form, L(x, ∂x) = aij(x)∂i∂i, for n ≥ 3. We assume that the coefficients have modulus of continuity satisfying the square Dini condition. For fixed y, we construct a solution of LZy(x) = 0 for 0 < |x −y | < ε with explicit leading order term which is O(|x−y | 2−n e I(x,y) ) as x → 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 → y, then we can solve L(x, ∂x)F(x, y) = δ(x − y), and F(x, y) is asymptotic as x → y to the fundamental solution for the constant coefficient operator L(y, ∂x). On the other hand, if I(x, y) → − ∞ as x → y then the solution Zy(x) violates the “extended maximum principle ” of Gilbarg & Serrin [8] and is a distributional solution of L(x, ∂x)Zy(x) = 0 for |x − y | < ε although Zy is not even bounded as x → y. 1.