A nonsymmetric coercive elliptic form

Given: The Axiom of Choice and Countable Choice; $n\ge 2$; a nonempty open $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ bounded in one direction; $b=e_1$; the form $a$ above; and the stated bump supported in a ball about $x_0$ with $x_{0,2}\mathrm{\ne }0$.

Since $b=e_1$ has $\mathrm{\parallel }b{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}=1$, Cauchy--Schwarz gives $\mathrm{\mid }a(u,v)\mathrm{\mid }\le \mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2+\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2\mathrm{\parallel }v{\mathrm{\parallel }}_2\le 2\mathrm{\parallel }u{\mathrm{\parallel }}_{H_0^1}\mathrm{\parallel }v{\mathrm{\parallel }}_{H_0^1}$; the form is bounded and sesquilinear (The elliptic form is well defined and bounded on $H^1$, Uniformly elliptic divergence-form operators and their sesquilinear forms, Holder's inequality for integrals, including the endpoint cases).

Poincar'e gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{H_0^1}^2\le (1+C_P^2)\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2^2$, so the principal form has coercivity constant $1\mathrm{/}(1+C_P^2)$ (Coercivity of the principal Dirichlet form, The Sobolev space $H^1$ is a Hilbert space, Integer-order Sobolev spaces and their norms).

For $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, its zero extension is smooth and compactly supported in ${\mathbb{R}}^n$. Choose $R>0$ so that its support lies in $(-R,R{)}^n$. For each fixed $x^{\mathrm{\prime }}=(x_2,\dots ,x_n)$, the function $t\mapsto \mathrm{\mid }\phi (t,x^{\mathrm{\prime }}){\mathrm{\mid }}^2$ has compact support in $(-R,R)$, so the one-dimensional fundamental theorem gives ${\int }_{-R}^R{\mathrm{\partial }}_1\mathrm{\mid }\phi (t,x^{\mathrm{\prime }}){\mathrm{\mid }}^{2}dt=0$. Fubini on the cube then gives ${\int }_{\mathrm{\Omega }}{\mathrm{\partial }}_1\mathrm{\mid }\phi {\mathrm{\mid }}^2=0$, hence $\mathrm{Re}{\int }_{\mathrm{\Omega }}{\mathrm{\partial }}_1\phi \bar{\phi }=0$. Also $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ is dense in $H_0^1(\mathrm{\Omega })$, and $u\mapsto {\int }_{\mathrm{\Omega }}{\mathrm{\partial }}_{1}u\bar{u}$ is continuous in the $H^1$ norm by Cauchy--Schwarz and [F1] (The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$, Tonelli and Fubini for the completed product, with only almost-everywhere section measurability, Test function space d of an open set, Zero-boundary Sobolev space as a norm closure, Holder's inequality for integrals, including the endpoint cases).

The coordinate product rule gives ${\mathrm{\partial }}_1(x_2\eta )=x_2{\mathrm{\partial }}_1\eta$ and ${\mathrm{\partial }}_1(x_1\eta )=\eta +x_1{\mathrm{\partial }}_1\eta$ (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$). For a radial bump about $x_0$, reflection $x_2\mapsto 2x_{0,2}-x_2$ leaves ${\eta }^2$ unchanged and has absolute Jacobian $1$; applying The published Riemann change-of-variables theorem already gives the Lebesgue formula for continuous compactly supported integrands on ${\mathbb{R}}^n$ to $(x_2-x_{0,2}){\eta }^2$ shows its integral equals its negative, hence is zero.

The standard smooth step $\sigma$ is smooth, takes values in $[0,1]$, vanishes for $t\le 0$ and equals $1$ for $t\ge 1$ (The standard smooth step function). Repeated coordinate chain and product rules give smoothness of its composition with a polynomial (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$, Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$).

The real or complex Hilbert space $H_0^1(\mathrm{\Omega })$ is complete, and Lax--Milgram applies to every bounded coercive sesquilinear form without symmetry; the energy-minimisation conclusion requires symmetry (The Sobolev space $H^1$ is a Hilbert space, The Lax--Milgram theorem, Symmetric Lax--Milgram is energy minimisation, Nonsymmetric Lax--Milgram is not a scalar minimisation principle, Weak Dirichlet solutions for a divergence-form operator).