Weak Dirichlet solutions for a divergence-form operator

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), and let $L$ and its form $a$ be as in Uniformly elliptic divergence-form operators and their sesquilinear forms on an open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$. Homogeneous problem. Given $F\in H^{-1}(\mathrm{\Omega })$ (The negative Sobolev space $H^{-1}(\mathrm{\Omega })$), a weak solution of $Lu=F$ with zero boundary values is a class $u\in H_0^1(\mathrm{\Omega })$ (Zero-boundary Sobolev space as a norm closure, Integer-order Sobolev spaces and their norms) with $a(u,v)=F(v)\text{for every }v\in H_0^1(\mathrm{\Omega })\mathrm{.}$ Inhomogeneous problem. Additionally assume the Axiom of Choice (The Axiom of Choice) for the trace supplier, and let $n\ge 2$ and $\mathrm{\Omega }$ be a bounded $C^1$ domain (Bounded C^k domains and boundary charts), let $g\in H^{1\mathrm{/}2}(\mathrm{\partial }\mathrm{\Omega }):=W^{1\mathrm{/}2,2}(\mathrm{\partial }\mathrm{\Omega })$ in the boundary scale of The fractional Sobolev space on a compact $C^1$ boundary and let $T$ be the trace operator of The $L^p$ trace operator on a bounded $C^1$ domain. A weak solution with boundary data $g$ is a class $u\in H^1(\mathrm{\Omega })$ with $Tu=g$ and $a(u,v)=F(v)$ for every $v\in H_0^1(\mathrm{\Omega })$. The defining identities are identities between functionals on the $H_0^1$ classes, so they are independent of the chosen almost-everywhere representatives of $u$, of the coefficients and of the data (Weak differentiation ignores null-set changes, Complex Lp classes and Euclidean test-function conventions); the boundary condition is imposed through the trace of The $L^p$ trace operator on a bounded $C^1$ domain, never by pointwise evaluation. The Dirichlet condition is imposed by $u\in H_0^1(\mathrm{\Omega })$ in the homogeneous problem and by $Tu=g$ in the inhomogeneous problem; testing against $v\in H_0^1$ expresses the weak equation and does not by itself impose boundary data; the integrals are the ones proved absolutely convergent in The elliptic form is well defined and bounded on $H^1$.