The formal adjoint and the adjoint weak Dirichlet problem

Assume Countable Choice. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open and let $L,a$ be as in Uniformly elliptic divergence-form operators and their sesquilinear forms. The adjoint form is $a^{\ast }(u,v):=\overline{a(v,u)},$ a bounded sesquilinear form on $H^1(\mathrm{\Omega })$ with the same bound as $a$ (Bounded, coercive and symmetric sesquilinear forms, The elliptic form is well defined and bounded on $H^1$); in coefficients $a^{\ast }(u,v)={\int }_{\mathrm{\Omega }}(\overline{a^{ji}}{D}_{j}u\overline{D_{i}v}+\overline{b^i}u\overline{D_{i}v}+\bar{c}u\bar{v})dx\mathrm{.}$ The formal adjoint is the expression $L^{\ast }u:=-D_i(\overline{a^{ji}}{D}_{j}u)-D_i(\overline{b^i}u)+\bar{c}u$, understood as a distribution: for $w,v\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, $⟨L^{\ast }w,\bar{v}⟩=a^{\ast }(w,v)$. Indeed the coefficient products are locally integrable, hence define regular distributions by Locally integrable functions embed in distributions, and the signed derivative rule of Distributional derivative gives exactly the displayed form. Even for smooth $w$, $L^{\ast }w$ need not be a locally integrable function when the coefficients are merely measurable; an integral $\int (L^{\ast }w)\bar{v}$ is used only when it is represented by such a function; the form $a^{\ast }$ is the primary object and is defined before any integration by parts. The adjoint weak Dirichlet problem with datum $f\in L^2(\mathrm{\Omega })$ asks for $v\in H_0^1(\mathrm{\Omega })$ with $a^{\ast }(v,w)=(f,w{)}_{L^2}\text{for every }w\in H_0^1(\mathrm{\Omega }),$ and its homogeneous version is $a^{\ast }(v,w)=0$ for all $w$. No orthogonality is invoked in this definition. Since $\mathrm{Re}{a}^{\ast }(u,u)=\mathrm{Re}a(u,u)$, the Garding constants of Garding's inequality for a divergence-form elliptic operator also apply to $a^{\ast }$, and $a_{\mu }^{\ast }:=a^{\ast }+\mu (\cdot ,\cdot {)}_{L^2}$ is coercive for $\mu \ge \beta$ (A sufficiently large shift is coercive).

Conventions recorded with the definition. All pairings are the $L^2$ or $H_0^1$ pairings of the cited items, with conjugation in the second slot; the datum $f\in L^2(\mathrm{\Omega })$ acts through the conjugate-linear functional $w\mapsto (f,w{)}_{L^2}$, which is an element of $H^{-1}(\mathrm{\Omega })$ by the Cauchy--Schwarz estimate $\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2}\le \mathrm{\parallel }w{\mathrm{\parallel }}_{H_0^1}$ (The negative Sobolev space $H^{-1}(\mathrm{\Omega })$, Integer-order Sobolev spaces and their norms, Zero-boundary Sobolev space as a norm closure, Real and imaginary parts, complex conjugation, and modulus). The formal expression $L^{\ast }$ is recorded as the operator whose weak pairing reproduces $a^{\ast }$ on smooth compactly supported functions; $L^{\ast }$ is not claimed to be of the same divergence form as $L$, and no boundary condition is attached to it beyond the test class $H_0^1(\mathrm{\Omega })$. The adjoint weak problem is stated for $H_0^1$ test functions exactly as in Weak Dirichlet solutions for a divergence-form operator, and no existence or uniqueness is asserted here.