Weak subsolutions and supersolutions of a divergence-form equation

Assume Countable Choice and the Axiom of Choice for the Sobolev, trace and embedding interfaces below. Let $n\ge 1$, let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be open, 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 $M_a,M_b,M_c$. For the order comparison below, take real coefficients and a real-valued source $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$ (Locally integrable functions as regular distributions, The space $L^p(\mu )$ as the quotient by null functions).

A real class $u\in H^1(\mathrm{\Omega };\mathbb{R})$ (The notation $H^k$ and the reserved zero-boundary symbol) is a local weak subsolution of $Lu=f$ on $\mathrm{\Omega }$ if $a(u,\phi )\le {\int }_{\mathrm{\Omega }}f\phi dx\text{for every nonnegative }\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R}),$ and a local weak supersolution if the reverse inequality holds; it is a local weak solution if equality holds for every real $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$. These tests make every pairing finite for $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$.

Global Sobolev-test version. If the source defines a continuous functional $F\in H^{-1}(\mathrm{\Omega }):=(H_0^1(\mathrm{\Omega }){)}^{\ast }$ (The negative Sobolev space $H^{-1}(\mathrm{\Omega })$), then a real $u\in H^1(\mathrm{\Omega })$ is a global weak subsolution if $a(u,v)\le F(v)\text{for every nonnegative }v\in H_0^1(\mathrm{\Omega }),$ with the reverse inequality defining a global weak supersolution and equality defining a global weak solution. The local and global formulations agree when both apply, by continuity of the form and $F$ and the following positive-cone density argument. Given $0\le v\in H_0^1$, choose real $z_j\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ converging to $v$ in $H^1$ and a subsequence converging a.e.; such a subsequence follows by choosing $\mathrm{\parallel }{z}_j-v{\mathrm{\parallel }}_2^2\le 2^{-3j}$ and applying Chebyshev and countable subadditivity to $\{\mathrm{\mid }{z}_j-v\mathrm{\mid }>2^{-j}\}$. The positive-part chain rule gives $D{z}_j^{+}-Dv={\mathbf{1}}_{\{z_j>0\}}(D{z}_j-Dv)+({\mathbf{1}}_{\{z_j>0\}}-{\mathbf{1}}_{\{v>0\}})Dv$. The second term tends to zero in $L^2$ by dominated convergence, since its indicator converges where $v>0$ and $Dv=0$ a.e. where $v=0$; the first term and the function difference converge in $L^2$. Thus $z_j^{+}\to v$ in $H^1$. Each $z_j^{+}$ has compact support, so zero extension followed by nonnegative unit-mass mollification with sufficiently small radius gives a nonnegative $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ approximant within $1\mathrm{/}j$ in $H^1$ (Positive, negative, and truncated Sobolev functions, Compactly supported Sobolev functions extend by zero in every integer order, Local smooth approximation in integer-order Sobolev spaces, Dominated convergence, Chebyshev-Markov inequality for the integral). In particular, $f\in L^2(\mathrm{\Omega })$ suffices by Cauchy--Schwarz and $H_0^1\hookrightarrow L^2$; on a bounded $C^1$ domain, $f\in L^q(\mathrm{\Omega })$ with $q>n\mathrm{/}2$ for $n\ge 3$ (or $q>1$ for $n=2$) suffices because $q^{\mathrm{\prime }}$ lies in the Sobolev range $H_0^1\hookrightarrow L^{q^{\mathrm{\prime }}}$ (The Sobolev inequality for zero-boundary Sobolev closures on open sets, The critical Sobolev embedding into every finite $L^q$, Holder's inequality for integrals, including the endpoint cases). If $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(\mathrm{\Omega })$, the local inequality also extends to nonnegative $H_0^1(U)$ tests on any bounded open $U⋐\mathrm{\Omega }$, since $f{\mathrm{\mid }}_U\in L^2(U)$.

For complex-valued coefficients or classes, only the weak-solution identity with a specified continuous complex source functional is used; no subsolution or supersolution order is defined by comparing complex numbers. In the real setting, a weak solution is both a subsolution and a supersolution exactly when the same source functional is used in both inequalities.

Signed essential extrema. For a real measurable class on a positive-measure set $E$, write ${\mathrm{ess\hspace{0.17em}sup}}_{E}u:=\inf \{t\in \mathbb{R}:u\le t\text{ a.e. on }E\}$ in the extended reals, and ${\mathrm{ess\hspace{0.17em}inf}}_{E}u:=-{\mathrm{ess\hspace{0.17em}sup}}_E(-u)$. A finite essential supremum $s$ is itself an a.e. upper bound: take the union of the null exceptional sets for the bounds $s+1\mathrm{/}j$. These signed extrema differ from the essential supremum of $\mathrm{\mid }u\mathrm{\mid }$ used to define the $L^{\mathrm{\infty }}$ norm. For a continuous representative on an open set, its pointwise and essential extrema agree, since a strict violation of an essential bound would hold on a nonempty open set of positive measure.

Weak boundary order (for $n\ge 2$). Let in addition $n\ge 2$ and let $\mathrm{\Omega }$ be a bounded $C^1$ domain (Bounded C^k domains and boundary charts) with trace operator $T$ (The $L^p$ trace operator on a bounded $C^1$ domain). For real $u,v\in H^1(\mathrm{\Omega })$ and $k\in \mathbb{R}$ one writes $u\le k$ on $\mathrm{\partial }\mathrm{\Omega }$ if $(u-k{)}^{+}\in H_0^1(\mathrm{\Omega })$, and $u\le v$ on $\mathrm{\partial }\mathrm{\Omega }$ if $(u-v{)}^{+}\in H_0^1(\mathrm{\Omega })$; by The kernel of the trace is the closure of the test functions these are respectively the statements $Tu\le k$ and $Tu\le Tv$ a.e. on $\mathrm{\partial }\mathrm{\Omega }$, as justified by A function whose trace is at most a level has positive part in the zero-boundary space ↗, and it is independent of the chosen representatives. The boundary supremum is ${\sup }_{\mathrm{\partial }\mathrm{\Omega }}u:=\inf \{k\in \mathbb{R}:u\le k\text{ on }\mathrm{\partial }\mathrm{\Omega }\}\in \mathbb{R}\cup \{+\mathrm{\infty }\},$ with $\inf \mathrm{\varnothing }:=+\mathrm{\infty }$; the set is nonempty as soon as $u$ is essentially bounded above.

• Sign convention. In the real order theory, the subsolution inequality is $a(u,v)\le {\int }_{\mathrm{\Omega }}fv$ against nonnegative tests. For the operator $L=-D_i(a^{ij}{D}_j)+b^i{D}_i+c$, the favourable pointwise sign in the maximum principle is $c\ge 0$; negating a supersolution preserves the same coefficients, so the same sign is favourable for the corresponding minimum estimate.
• Real order versus complex identities. The maximum-principle, De Giorgi and Harnack results use real-valued $u$, real coefficients and real sources, so their inequalities compare real numbers. Complex local weak solutions use the compactly supported identity of Local weak solutions of a divergence-form operator; a complex global weak identity uses a specified continuous functional on $H_0^1$. No order notion is assigned to a complex-valued form.
• No boundary condition is imposed by the subsolution or supersolution notion itself, and the boundary order is only introduced on a bounded $C^1$ domain, through the trace; it is never read off pointwise boundary values of a class.

• Simon, Lectures on Partial Differential Equations, Lecture 13, printed pp. 147-158: the real weak form against nonnegative $\phi \in C_c^{\mathrm{\infty }}$, the conventions (i)-(iv) for $u\le 0$ on $\mathrm{\partial }\mathrm{\Omega }$ and ${\sup }_{\mathrm{\partial }\mathrm{\Omega }}u=\inf \{k:u\le k\text{ on }\mathrm{\partial }\mathrm{\Omega }\}$, and the weak maximum principle Theorem 4. Simon works with real-valued data; the present definition records the local real order convention and the separate $H^{-1}$ global extension.
• Teschl, PDE: From Classical to Modern, Chapter 10 Section 1: Theorem 10.1, Lemma 10.2 and the same boundary convention $(v-u{)}^{+}\in H_0^1(U)$ for $v\le u$ on $\mathrm{\partial }U$.
• Schikorra, Partial Differential Equations, Chapter 2 Sections II.1-II.2: Definitions II.1.1 and II.1.3, the sign convention for the zeroth-order term, and Theorem II.2.1.