 Solution

Write $H^1=W^{1,2}$ and $H_0^1=W_0^{1,2}$. The Dirichlet boundary condition in the Sobolev space sense means $u-\psi \in H_0^1(\Omega )$; this definition remains meaningful even without regularity assumptions on the boundary. For bounded full coefficients, a weak solution is a function satisfyingwhereThe sign follows by integration by parts in $D_i(a_{ij}{D}_{j}u)+qu=f$. No classical derivatives of the measurable coefficients are taken. Equivalently one may first test against $C_c^{\infty }(\Omega )$ and extend by density, provided this bilinear form is bounded on $H^1\times H_0^1$. All integrals in this definition must exist: as explained below, the printed quadratic bounds alone do not guarantee this when $A=(a_{ij})$ is nonsymmetric.