Abstract:

In this paper we give the existence of weak solutions to a second order elliptic partial differential equation with square integrable regularity of leading coefficients.Specifically,we consider the elliptic equation - ∂ _j(a_ij(x) ∂_iu)=f⁰ + ∂_ifⁱ inΩ,u =0on the boundary,with a_ij =a_ji，a_ij uniform elliptic,and a_ij ∈L²(Ω).First,we approximate a_ijby a^(m)_ij which belongs to L∞(Ω),and then prove the existence theorem by applying the well known results concerning solvability of elliptic partial differential equation together with the standard energy method.

Key words:  , Elliptic equation, Existence, Energy method