Local weak solutions of a divergence-form operator

Assume Countable Choice for the Sobolev interfaces. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open and not necessarily bounded, $n\ge 1$, let $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, and let $L$ and its sesquilinear form $a$ be as in Uniformly elliptic divergence-form operators and their sesquilinear forms, with ellipticity constant $\theta$ and coefficient bounds $M_a,M_b,M_c$. Let $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(\mathrm{\Omega })$ (The space $L^p(\mu )$ as the quotient by null functions).

A class $u\in H^1(\mathrm{\Omega };\mathbb{K})$ (The notation $H^k$ and the reserved zero-boundary symbol) is a local weak solution of $Lu=f$ on $\mathrm{\Omega }$ if $a(u,v)={\int }_{\mathrm{\Omega }}f\bar{v}dx\text{for every }v\in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{K}),$ where $a$ is the sesquilinear form of Uniformly elliptic divergence-form operators and their sesquilinear forms and the right-hand side is finite because $v$ is bounded with compact support in $\mathrm{\Omega }$ and $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(\mathrm{\Omega })$.

Equivalences and well-definedness. Write ${\mathrm{\Omega }}_2⋐\mathrm{\Omega }$ when ${\mathrm{\Omega }}_2$ is open, bounded and $\overline{{\mathrm{\Omega }}_2}\subseteq \mathrm{\Omega }$. For such an ${\mathrm{\Omega }}_2$ and $v\in H_0^1({\mathrm{\Omega }}_2)$, regard $v$ as its zero extension to $\mathrm{\Omega }$. This extension is in $H^1(\mathrm{\Omega })$ with the same norm: extend a defining sequence in $C_c^{\mathrm{\infty }}({\mathrm{\Omega }}_2)$ by zero; its function and gradient sequences converge in $L^2(\mathrm{\Omega })$, and passing the compact-test identity to the limit identifies the extended gradient. Thus $a(u,v)$ and ${\int }_{{\mathrm{\Omega }}_2}f\bar{v}$ are finite by The elliptic form is well defined and bounded on $H^1$ and Cauchy-Schwarz. The defining identity for all $v\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ is equivalent to the identity $a(u,v)={\int }_{{\mathrm{\Omega }}_2}f\bar{v}dx\text{for every bounded open }{\mathrm{\Omega }}_2⋐\mathrm{\Omega }\text{ and every }v\in H_0^1({\mathrm{\Omega }}_2),$ because $C_c^{\mathrm{\infty }}({\mathrm{\Omega }}_2)$ is dense in $H_0^1({\mathrm{\Omega }}_2)$ by definition of the closure (Zero-boundary Sobolev space as a norm closure) and both sides are continuous in $v$ in the $H^1(\mathrm{\Omega })$ norm: $a$ is bounded on $H^1(\mathrm{\Omega })$ by The elliptic form is well defined and bounded on $H^1$, and $\mathrm{\mid }{\int }_{{\mathrm{\Omega }}_2}f\bar{v}dx\mathrm{\mid }\le \mathrm{\parallel }f{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}_2)}\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}_2)}$ by Cauchy-Schwarz, while the $H^1({\mathrm{\Omega }}_2)$ norm controls the $L^2({\mathrm{\Omega }}_2)$ norm. If in addition $f\in L^2(\mathrm{\Omega })$ then the defining identity is equivalent to $a(u,v)={\int }_{\mathrm{\Omega }}f\bar{v}dx$ for every $v\in H_0^1(\mathrm{\Omega })$, since $v\mapsto a(u,v)-{\int }_{\mathrm{\Omega }}f\bar{v}$ is then bounded on the whole space $H_0^1(\mathrm{\Omega })$ and $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ is dense in it. The definition depends on $u$, on the coefficients and on $f$ only through their almost-everywhere classes; this is the class-level statement of The elliptic form is well defined and bounded on $H^1$. No boundary condition is imposed. For a fixed datum $f\in L^2(\mathrm{\Omega })$, the zero-boundary Dirichlet notion of Weak Dirichlet solutions for a divergence-form operator is exactly this local weak equation together with $u\in H_0^1(\mathrm{\Omega })$. Without restricting the data class, the two notions are not ordered: Dirichlet data may be arbitrary elements of the dual of $H_0^1$, whereas this definition requires an $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$ representative.

Locality. If ${\mathrm{\Omega }}_0\subseteq \mathrm{\Omega }$ is open and $u$ is a local weak solution of $Lu=f$ on $\mathrm{\Omega }$, then the restriction $u{\mathrm{\mid }}_{{\mathrm{\Omega }}_0}$ is a local weak solution of $Lu=f{\mathrm{\mid }}_{{\mathrm{\Omega }}_0}$ on ${\mathrm{\Omega }}_0$ with the same coefficient functions restricted to ${\mathrm{\Omega }}_0$: every test function $\phi \in C_c^{\mathrm{\infty }}({\mathrm{\Omega }}_0)$ extends by zero to a test function of $\mathrm{\Omega }$, and the defining integrals over $\mathrm{\Omega }$ are the integrals over ${\mathrm{\Omega }}_0$ because $\phi$ and all its derivatives vanish outside ${\mathrm{\Omega }}_0$. The equation is therefore a local condition, which is why every regularity argument below may be localised to a ball, a half-ball or a chart without changing the coefficients or the datum.

The $H^1$ versus $H_0^1$ convention. The solution is required to lie in $H^1(\mathrm{\Omega })$ and the tests are required to vanish near $\mathrm{\partial }\mathrm{\Omega }$; this is the interior formulation used in the regularity proof. Here $H_0^1({\mathrm{\Omega }}_2)$ for a bounded ${\mathrm{\Omega }}_2⋐\mathrm{\Omega }$ is the closure of $C_c^{\mathrm{\infty }}({\mathrm{\Omega }}_2)$ in the $H^1({\mathrm{\Omega }}_2)$ norm (Zero-boundary Sobolev space as a norm closure, Complex Lp classes and Euclidean test-function conventions), and all the integrals are read in the class conventions of The space $L^p(\mu )$ as the quotient by null functions.