Complex polynomial weak derivatives and zero boundary preservation

By the weak test identity in [F1], taking complex conjugates shows $D_j\bar{u}=\overline{D_{j}u}$: the test functions may be taken real, and the identity then extends by linearity to complex tests. The same applies to $v$. Repeatedly apply [F3] to the bounded factors $u$ and $\bar{u}$; every intermediate positive-degree product is bounded and lies in $H^1$. This gives the displayed formula for $D_{j}g(u)$ and $g(u)\in H^1$. The value and derivative satisfy $\mathrm{\mid }g(u)\mathrm{\mid }\le \mathrm{\mid }\lambda \mathrm{\mid }\mathrm{\mid }u{\mathrm{\mid }}^m$ and $\mathrm{\mid }{D}_{j}g(u)\mathrm{\mid }\le m\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{\mid }u{\mathrm{\mid }}^{m-1}\mathrm{\mid }{D}_{j}u\mathrm{\mid }$. Taking $L^2$ norms proves the first estimate.

Suppose now $u\in H_0^1(\mathrm{\Omega })\cap L^{\mathrm{\infty }}(\mathrm{\Omega })$. By [F2] choose ${\varphi }_k\in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{C})$ converging to $u$ in $H^1$. Choose $R>\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}+1$, take the cutoff $\chi$ from [F7], and put $T(t)=t\chi (t\mathrm{/}R)$. Thus $T$ is smooth and globally Lipschitz, $T(0)=0$, $T(t)=t$ and $T^{\mathrm{\prime }}(t)=1$ for $\mathrm{\mid }t\mathrm{\mid }\le R$, and both $T$ and $T^{\mathrm{\prime }}$ are bounded. Define $w_k=T(\mathrm{Re}{\varphi }_k)+iT(\mathrm{Im}{\varphi }_k)$. Each $w_k$ is smooth and compactly supported in $\mathrm{\Omega }$, and the sequence is uniformly bounded in $L^{\mathrm{\infty }}$. By [F5], after taking a subsequence, both components of ${\varphi }_k$ converge almost everywhere to the corresponding components of $u$. Since $T$ is Lipschitz and fixes these components almost everywhere, $w_k\to u$ in $L^2$. For either real component $q$ of $u$ and corresponding $q_k$ of ${\varphi }_k$, [F4] gives $D_{j}T(q_k)=T^{\mathrm{\prime }}(q_k)D_j{q}_k$ and $D_{j}T(q)=D_{j}q$. The difference is $T^{\mathrm{\prime }}(q_k)(D_j{q}_k-D_{j}q)+(T^{\mathrm{\prime }}(q_k)-1)D_{j}q$. The first term tends to zero in $L^2$; the square of the second tends to zero almost everywhere and is dominated by a constant times $\mathrm{\mid }{D}_{j}q{\mathrm{\mid }}^2$, so [F6] makes its $L^2$ norm tend to zero. Consequently $w_k\to u$ in $H^1$.

Put $M=\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\infty }}$ and $h=v-u$. Telescoping the $m$ factors in $v^a{\bar{v}}^b-u^a{\bar{u}}^b$ gives $\mathrm{\mid }g(v)-g(u)\mathrm{\mid }\le m\mathrm{\mid }\lambda \mathrm{\mid }{M}^{m-1}\mathrm{\mid }h\mathrm{\mid }$. The coefficients of $D_{j}v$ and $\overline{D_{j}v}$ in step 1.1 are polynomials of degree $m-1$ in $v,\bar{v}$; telescoping their factors gives coefficient differences bounded by $C_m\mathrm{\mid }\lambda \mathrm{\mid }{M}^{m-2}\mathrm{\mid }h\mathrm{\mid }$. Subtract the derivative formulas using $D_{j}v-D_{j}u=D_{j}h$: $\mathrm{\mid }{D}_{j}g(v)-D_{j}g(u)\mathrm{\mid }\le C_m\mathrm{\mid }\lambda \mathrm{\mid }(M^{m-1}\mathrm{\mid }{D}_{j}h\mathrm{\mid }+M^{m-2}\mathrm{\mid }h\mathrm{\mid }\mathrm{\mid }{D}_{j}u\mathrm{\mid })$. Taking $L^2$ norms, using essential suprema for $u,v,h$, and bounding $M^r$ by a constant depending only on $r$ times $\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}^r+\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\infty }}^r$ for $r=m-1,m-2$, yields the stated symmetric estimate after enlarging $C_m$.

Each $g(w_k)$ lies in $C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{C})$ because $g$ is a polynomial with $g(0)=0$. The uniform bound on $w_k$ and the almost-everywhere convergence from step 1.2 imply $g(w_k)\to g(u)$ in $L^2$ by the pointwise polynomial difference bound. In the derivative formula of step 1.1, subtract and add the coefficients at $w_k$ multiplying $D_{j}u$: the terms containing $D_j{w}_k-D_{j}u$ tend to zero in $L^2$ by the uniform coefficient bounds and step 1.2. Each remaining coefficient difference tends to zero almost everywhere, is uniformly bounded, and multiplies $D_{j}u$ or its conjugate; [F6] applied to its squared modulus gives convergence to zero in $L^2$. Hence $g(w_k)\to g(u)$ in $H^1$, and [F2] proves $g(u)\in H_0^1(\mathrm{\Omega })$. This uses no boundary regularity or trace characterization.