Boundary energy, local inverses, and harmonic data

A boundary condition plays two distinct roles. It selects an energy space before a solution has pointwise values, and it removes one of the normal modes in a local solver. We develop both roles and connect them through a regularity argument that starts with only one derivative. The harmonic case then turns interior solutions into operators acting on boundary data.

The order of construction is intentional: weak existence does not depend on boundary smoothness; regularity does. A separate normal-mode calculation records the precise extension from real to complex quadratic principal symbols. Higher-order systems and arbitrary boundary conditions require their own complementing-condition theory.

1. Spaces, conventions, and exact entry contracts

Put $D_j=-i\partial_j$, and use $\widehat u(\xi)=\int e^{-ix\cdot\xi}u(x)\,dx$ with inverse factor $(2\pi)^{-n}$. Thus $\sum D_j^2=-\Delta$. Inner products are linear in the first variable. On $\mathbb R^d$, $$\|u\|_{H^s}^2=(2\pi)^{-d}\int (1+|\xi|^2)^s|\widehat u(\xi)|^2\,d\xi. \tag{B1}$$ For an integer $k\geq0$, $H^k(V)$ denotes the space of $L^2$ functions with all weak derivatives through order $k$ in $L^2(V)$. On a half-space it is the restriction of the whole-space space, with equivalent norm; a proof appears below. For nonintegral $s$, the restriction space carries the quotient norm. We write $\gamma u$ for the boundary trace and $\nu$ for the exterior unit normal. An equation with Lipschitz leading coefficients and an initially $H^1$ solution uses the weak multiplication rule from Section 7 of Local inverses and distance-weighted elliptic estimates: $aD_jv=D_j(av)-(D_ja)v$.

The next four groups of facts identify exactly what the arguments use.

• Fourier and Sobolev facts. Fourier inversion and Plancherel extend to all of $L^2$, with the normalization (B1); Fourier transformation commutes with weak derivatives. The full-space extension is the initial Plancherel paragraph in the proof in Section 5 of Two measuring scales, one Weyl product (no selfadjoint-generator result from that item is needed). Its argument, from the Schwartz identity in Section 2 of Fourier transforms, finite spectra and convex separation, is: an isometry on a dense subspace extends to its completion, and its dense image is closed and therefore all of $L^2$. The same argument gives partial Fourier transformation with an additional $L^2$ variable, first on finite sums of product functions and then by density. Weighted completions define (B1). No pointwise Fourier transform is required for a general $L^2$ function.
• Weak derivatives and localization. The integration, weak-derivative, smooth approximation, and Lipschitz product results in Local inverses and distance-weighted elliptic estimates are used with $p=2$. Smooth changes of variables on compact coordinate patches, finite partitions of unity, and finite-dimensional linear algebra are entry facts. We supply below the precise Sobolev consequences of boundary flattening that are needed here.
• Local elliptic inverses. Section 2 of Local inverses and distance-weighted elliptic estimates supplies a fundamental solution for any homogeneous elliptic scalar polynomial, local bounds for convolution through its order, and the two convolution inverse identities on their stated domains. Section 7 of Local inverses and distance-weighted elliptic estimates supplies the Lipschitz commutator, and Section 8 of Local inverses and distance-weighted elliptic estimates supplies interior $H^2$ regularity from an $H^1$ solution with $L^2$ principal part. Our boundary argument proves the additional trace and reflection assertions instead of treating the interior theorem as a boundary theorem.
• Boundary operator calculus. Only the layer-potential portion uses the order $-1$ operators, local quantization, Sobolev mapping, and smooth compact-manifold assembly proved in Symbols, operators and Sobolev scales and Detecting regularity without choosing coordinates, and compact-perturbation Fredholm theory in Finite defects under perturbation. The particular kernels, jumps, and boundary data spaces must still be checked below.

All Sobolev estimates are complex-linear unless an energy argument explicitly assumes a real symmetric positive coefficient matrix. Bounds may depend on a fixed coordinate patch and coefficient bounds. A local inverse makes no assertion of uniqueness without conditions on the rest of the boundary.

2. Traces, lifts, and a boundary-free energy domain

Write $x=(z,t)\in\mathbb R^{n-1}\times(0,\infty)$, $\lambda(\zeta)=(1+|\zeta|^2)^{1/2}$. The following elementary estimate is the source of the half derivative at the boundary. For a smooth scalar function $h$ decreasing at infinity, $$\lambda |h(t)|^2 \leq\int_t^\infty\bigl(\lambda^2|h(r)|^2+|h'(r)|^2\bigr)\,dr. \tag{B2}$$ Indeed integrate $-(|h|^2)'$ and use $2\lambda |h||h'|\leq\lambda^2|h|^2+|h'|^2$. The same inequality holds for an $H^1$ function on the ray by approximation, or by choosing a sequence tending to infinity on which its value tends to zero. Integrating (B2) in $\zeta$ proves $$\sup_{t\geq0}\|u(\cdot,t)\|_{H^{1/2}}\leq C\|u\|_{H^1(\mathbb R^n_+)}. \tag{B3}$$ Here and below the assertion about slices refers to their continuous representative, not to an arbitrary representative on sets of measure zero.

We justify the approximation and the weak-space interpretation. If $u,\partial_tu\in L^2$, testing against products of a test function in $z$ and one in $t$ gives the one-dimensional weak derivative identity for almost every tangential slice. A one-dimensional function with integrable weak derivative equals an absolutely continuous primitive plus a constant: subtract its integral primitive; the remaining distribution has derivative zero, as follows by testing against the derivative of every compactly supported test function of integral zero. Reflecting this representative evenly across zero therefore gives an $H^1$ function. Its tangential derivatives are even reflections and its normal derivative is the signed reflection of $\partial_tu$; integration by parts on the two rays cancels the boundary terms. Fubini yields a bounded even extension on $H^1$. Whole-space convolution and cutoffs now give approximation by restrictions of $C_c^\infty(\mathbb R^n)$. This proves both the weak-derivative characterization of the restriction space and the density used in (B3). The case $n=1$ has no tangential variable and uses exactly the same one-dimensional argument.

For smooth approximants the map $t\mapsto u(\cdot,t)$ is continuous into $H^{1/2}$. Bound (B3) makes these maps uniformly Cauchy on the closed ray, proving bounded continuity for every $H^1$ function. In particular $\gamma:H^1(\mathbb R^n_+)\to H^{1/2}(\mathbb R^{n-1})$ is bounded. More generally, for every integer $k\geq1$, $$\gamma:H^k(\mathbb R^n_+)\longrightarrow H^{k-1/2}(\mathbb R^{n-1}), \qquad \widehat{L\varphi}(\zeta,t)=e^{-t\lambda(\zeta)}\widehat\varphi(\zeta) \tag{B4}$$ are bounded maps with $\gamma L=I$. To prove the trace bound, apply (B2) after multiplying by $\lambda^{2k-2}$; the two resulting integrals are controlled by the tangential derivatives of orders at most $k$ and their first normal derivatives. For the lift, each derivative of total order $j\leq k$ has squared integral at most $C\int\lambda^{2j-1}|\widehat\varphi|^2$, because $\int_0^\infty e^{-2t\lambda}dt=(2\lambda)^{-1}$. This proves (B4), including its inhomogeneous low frequencies.

For completeness, the restriction and weak norms agree at every integer $k$. On the negative ray set $Eu(z,t)=\sum_{j=1}^{k}c_j u(z,-jt)$, choosing the unique solution of $\sum_{j=1}^{k}c_j(-j)^\ell=1$, $0\leq\ell<k$. The matrix is Vandermonde. Derivatives through order $k-1$ match at zero; repeated integration by parts introduces no interface delta through order $k$. Change of variable on each reflected term bounds the extension in $H^k$. Smooth approximation follows by this extension and convolution. In particular tangential differentiation commutes with the trace whenever the indicated derivatives lie in $H^1$.

Let $u^0$ be extension by zero. Integration by parts, first on smooth approximants and then using (B3), gives the distribution identities $$\partial_t u^0=(\partial_tu)^0+\gamma u\otimes\delta_0, \qquad \partial_{z_j}u^0=(\partial_{z_j}u)^0. \tag{B5}$$ Consequently $u^0\in H^1(\mathbb R^n)$ if and only if $\gamma u=0$. For necessity, an $L^2$ function cannot be a nonzero distribution supported on a hyperplane: that hyperplane has Lebesgue measure zero. Conversely (B5) lists all its first derivatives as $L^2$ functions. These formulas also show why a nonzero boundary value cannot simply be extended by zero in an energy argument.

For any bounded open $\Omega\subset\mathbb R^n$, define $$V_\Omega=\overline{C_c^\infty(\Omega)}^{\ H^1(\mathbb R^n)}, \tag{B6}$$ identifying its elements with their restrictions to $\Omega$. Every such element vanishes almost everywhere outside $\Omega$. This is the homogeneous Dirichlet domain, even when the boundary is irregular. Multiplication by a smooth function with bounded value and first derivatives preserves it, by the product estimate and approximation.

If $\partial\Omega$ is $C^1$, three descriptions agree: (B6); functions in $H^1(\Omega)$ with zero local flattened trace; and $H^1(\mathbb R^n)$ functions supported in $\overline\Omega$. Here is the local-to-global proof. A $C^1$ flattening and its inverse have bounded first derivatives and Jacobians on a smaller compact patch. The chain rule and change of variables bound composition in $H^1$, initially for smooth functions and then for weak functions by approximation. A zero-trace function extends by zero across the flattened plane by (B5). Translate that extension a distance $\varepsilon$ into the upper half-space and convolve at radius less than $\varepsilon/2$; after a fixed smaller cutoff this gives smooth functions supported strictly on the interior side, converging in $H^1$. Pulling them back gives compactly supported $C^1$ functions in $\Omega$; ordinary convolution at a distance smaller than their distance from the boundary approximates each by $C_c^\infty(\Omega)$. A finite partition of unity and interior convolution prove inclusion in (B6). Conversely (B3) and the composition estimate give zero trace for limits of tests. Finally a whole-space $H^1$ function supported in $\overline\Omega$ has identical traces from both sides of each flattened plane, by the slice continuity proof; its exterior trace is zero and so is its interior trace. A $C^1$ boundary has measure zero, since finitely many graph patches have measure zero by Fubini. Thus there is no additional $H^1$ function carried only by the boundary. No support characterization for an arbitrary irregular domain has been used.

Inhomogeneous Dirichlet data is removed by a lift, not by a pointwise substitution without a function-space bound. If $\varphi\in H^{3/2}$ on a flat boundary and $\Phi=L\varphi\in H^2$, then $$Pu=f,\quad\gamma u=\varphi \quad\Longleftrightarrow\quad P(u-\Phi)=f-P\Phi,\quad\gamma(u-\Phi)=0. \tag{B7}$$ Bounded coefficients suffice for $P\Phi\in L^2$. For a divergence-form energy equation the corresponding lift is $H^{1/2}\to H^1$, and the adjusted forcing is a continuous functional on $V_\Omega$, generally in $H^{-1}$, not necessarily $L^2$. Smooth compact-boundary lifts are obtained from (B4) in charts and a partition of unity. At regularity $C^{1,1}$, meaning that the first chart derivatives are Lipschitz, the same $H^{3/2}\to H^2$ assertion holds: integer $H^2$ composition uses the bounded weak second derivatives of the charts, while the boundary $H^{3/2}$ norm is $H^1$ plus the half-order norm of first derivatives. That half-order norm is equivalent to $\iint |v(x)-v(y)|^2/|x-y|^{d+1}\,dx\,dy$, as Fourier transformation and the substitution $h=x-y$ show. Bi-Lipschitz changes of variables preserve this integral up to constants, and multiplication by a bounded Lipschitz function is bounded in it after separating differences and integrating near and away from the diagonal. These facts justify the stated chart norm without assuming a smooth boundary at this step.

3. The energy solution before boundary regularity

Let $\Omega$ be bounded and open. Assume $A=(a_{jk})$ is real, symmetric, continuous on $\overline\Omega$, and positive definite at every point there. Compactness supplies constants $0<\lambda\leq\Lambda<\infty$ with $$\lambda|\xi|^2\leq \sum a_{jk}(x)\xi_k\overline{\xi_j} \leq\Lambda|\xi|^2\quad(x\in\overline\Omega,\ \xi\in\mathbb C^n). \tag{B8}$$ For complex $\xi$, apply the real inequality to real and imaginary parts; symmetry cancels their imaginary cross terms. Define $$Q(u,v)=\int_\Omega\sum a_{jk}D_ku\,\overline{D_jv}, \qquad P=\sum_{j,k}D_j(a_{jk}D_k). \tag{B9}$$ Then for every $f\in L^2(\Omega)$ there is exactly one $u\in V_\Omega$ satisfying $$Q(u,v)=(f,v)\quad(v\in V_\Omega),\qquad \|u\|_{H^1(\Omega)}\leq C\|f\|_2. \tag{B10}$$ The equation holds distributionally, and no boundary regularity is assumed.

Proof. Translate coordinates so that $|x_1|\leq R$ on $\Omega$. For a compactly supported test function, integrate $\partial_1(x_1|u|^2)$ over the whole space. Cauchy-Schwarz gives $$\|u\|_2^2\leq2R\|u\|_2\|\partial_1u\|_2, \qquad \|u\|_2\leq2R\|\partial_1u\|_2. \tag{B11}$$ Equivalently, before bounding $|x_1|$, one has $\|u\|_2^2\leq4\|x_1\partial_1u\|_2^2$. Density extends these bounds to $V_\Omega$. Hence $Q(u,u)^{1/2}$ is a norm equivalent to its complete $H^1$ norm, and $|(f,v)|\leq C\|f\|_2 Q(v,v)^{1/2}$.

We give the minimization argument, including existence, instead of leaving a representation theorem implicit. Set $J(v)=Q(v,v)-2\operatorname{Re}(f,v)$. Its infimum $m$ is finite because $r^2-2Cr\|f\|_2$ is bounded below. A minimizing sequence $v_j$ is bounded in the $Q$ norm. The parallelogram identity gives $$Q(v_j-v_k,v_j-v_k) =2J(v_j)+2J(v_k)-4J((v_j+v_k)/2) \leq2J(v_j)+2J(v_k)-4m. \tag{B12}$$ It is therefore Cauchy and converges in $V_\Omega$ to a minimizer $u$. For real $t$, differentiating the quadratic polynomial $J(u+tv)$ at zero gives the real part of (B10). Applying it to $iv$ gives the imaginary part. Set $v=u$ and use the functional bound to obtain $Q(u,u)^{1/2}\leq C\|f\|_2$, hence (B10). The difference of two solutions has zero $Q$ norm. Testing (B10) on $C_c^\infty$ is exactly the distributional equation. Conversely that equation for an element of $V_\Omega$ extends to every $v\in V_\Omega$ by density and the continuity of both sides. This proves the equivalence as well as uniqueness. $\square$

The proof also applies to any continuous conjugate-linear forcing functional on $V_\Omega$, with its dual norm replacing $\|f\|_2$. Thus (B7) gives weak existence for lifted boundary values on a regular boundary, and a different lift gives the same solution because their difference belongs to $V_\Omega$.

The complete dual norm and measurable coefficients. The same proof needs only an essentially bounded measurable real symmetric matrix satisfying every bound in (B8) almost everywhere, with the stated constants $\lambda,\Lambda$. Pointwise real symmetry gives the complex inequality; diagonalizing the actual finite symmetric matrix shows $|A(x)\xi|\leq\Lambda|\xi|$. Thus $|Q(u,v)|\leq\Lambda\|\nabla u\|_2\|\nabla v\|_2$, and $\lambda\|\nabla v\|_2^2\leq Q(v,v)$. These estimates, (B11), completeness of (B6), and the exact identity (B12) prove existence and uniqueness for every bounded conjugate-linear $F$ by replacing $(f,v)$ by $F(v)$ in the entire minimization argument. This extends weak existence; the later regularity theorems retain their additional coefficient hypotheses.

Retain both norms of the forcing on this same space. Set $C_R=(1+4R^2)^{1/2}$, using the actual coordinate bound in (B11). Its proof gives $\|\nabla v\|_2\leq\|v\|_{H^1}\leq C_R\|\nabla v\|_2$. The two unit-ball inclusions and homogeneity then give $$\begin{split} \|F\|_{1,*}&=\sup_{\|v\|_{H^1}\leq1}|F(v)|,\qquad \|F\|_{\nabla,*}=\sup_{\|\nabla v\|_2\leq1}|F(v)|,\\ \|F\|_{1,*}&\leq\|F\|_{\nabla,*}\leq C_R\|F\|_{1,*}. \end{split} \tag{BO7}$$ Testing $Q(u,u)=F(u)$ proves, for $u\ne0$, $\lambda\|\nabla u\|_2^2\leq\|F\|_{\nabla,*}\|\nabla u\|_2$; division is legitimate by (B11). For $u=0$ the same bounds hold directly. Consequently the full constants are $$\|\nabla u\|_2\leq\lambda^{-1}\|F\|_{\nabla,*} \leq\lambda^{-1}C_R\|F\|_{1,*},\qquad \|u\|_{H^1}\leq\lambda^{-1}C_R^2\|F\|_{1,*}. \tag{BO8}$$ For an actual $H^1$ lift $\Phi$, the forcing for $w=u-\Phi$ is exactly $F(v)-Q(\Phi,v)$. It is bounded by $(\|F\|_{1,*}+\Lambda\|\nabla\Phi\|_2)\|v\|_{H^1}$. If another lift is $\Psi=\Phi+\chi$, with $\chi\in V_\Omega$, its solution is $w_\Psi=w_\Phi-\chi$, since substitution gives $Q(w_\Psi,v)=F(v)-Q(\Psi,v)$. Adding the respective lifts proves equality of the resulting original $u$.

Sergio Vessella’s freely accessible original-author notes, arXiv:2305.04765v1, give the variational construction in Chapter 4, Sections 4.1–4.3. The exact comparison in Dirichlet lifting and the full dual norm, (VE1)–(VE26), proves the isometry of its closure space with (B6), compares its original constants with (B8), and supplies complete counterexamples for the printed lifting sign and the printed estimate stab when its specified full dual norm is used. The source correction is identified there as editorial material. Here (B7) and (BO7)–(BO8) retain the negative lifted form and the required norm factors.

Example: an elliptic coefficient without classical first derivatives everywhere. In a ball take $a_{jk}(x)=(2+|x_1|)\delta_{jk}$. It is continuous, Lipschitz, and uniformly positive. Formula (B10) defines a unique energy solution. Expanding its operator produces the first-order coefficient $-i\,\operatorname{sgn}(x_1)$ in front of $D_1u$, defined almost everywhere. A theorem requiring continuous lower-order coefficients would not apply to this expansion. The bounded-lower-coefficient regularity theorem below does.

4. A reflected solver and its two inverse identities

Let $p_{\mathrm{orig}}$ be a real homogeneous elliptic quadratic polynomial, and write $p$ for its positive signed comparison. Its quadratic form is definite: on the connected unit sphere for $n\geq2$, a nonzero continuous real function has constant sign; for $n=1$ the assertion is immediate. Choose $\sigma\in\{1,-1\}$ so that $p=\sigma p_{\mathrm{orig}}$ is positive. It can be reduced to $|\eta|^2$ by a linear change preserving the plane $t=0$ and its positive side. Indeed write $p(\zeta,\tau)=\zeta^TB\zeta+2\tau c^T\zeta+a\tau^2$, where $a>0$. Completing the square leaves the positive Schur complement $B-cc^T/a$. A tangential shear removes $c$; a positive normal dilation and an invertible tangential change normalize the remaining blocks. The dual physical transformation has last coordinate a positive multiple of $t$. All derivative and measure norms are equivalent under this fixed change. The reflected calculation below uses coordinates in which $p(D)=-\Delta$. The exact signed comparison with the original form follows.

Here is the exact map back to the original form, including its sign and volume factor. Retain the original $p_{\mathrm{orig}}$, its sign $\sigma$, the positive comparison $p$, and the displayed $a,B,c$. Put $$S=B-\frac{cc^T}{a},\qquad F(w,\rho)=\left(S^{1/2}w+\frac{c}{\sqrt a}\rho,\ \sqrt a\,\rho\right), \qquad J_F=\sqrt a\,(\det S)^{1/2}.$$ The matrix $S$ is positive definite: for nonzero $\zeta$, evaluate the positive form at $\tau=-a^{-1}c^T\zeta$ to get $\zeta^TS\zeta>0$. The exact square and transformed full symbol are $$p(\zeta,\tau)=\zeta^TS\zeta+a(\tau+a^{-1}c^T\zeta)^2,\qquad \sigma p_{\mathrm{orig}}\!\left(S^{-1/2}\eta,\ a^{-1/2}\theta-a^{-1}c^TS^{-1/2}\eta\right) =|\eta|^2+\theta^2.$$ For $n=1$, $S$ is the empty matrix and $\det S=1$. The inverse coordinates are $w=S^{-1/2}(z-ct/a)$ and $\rho=t/\sqrt a$. Thus $F$ carries the upper half-space onto itself and the boundary plane onto itself. The chain rule gives $$(p_{\mathrm{orig}}(D_x)u)\circ F =\sigma\left[-\Delta_{w,\rho}(u\circ F)\right].$$ Indeed the exact derivative maps are $D_z=S^{-1/2}D_w$ and $D_t=a^{-1/2}D_\rho-a^{-1}c^TS^{-1/2}D_w$. Substituting both into the displayed full square gives $D_w^TD_w+D_\rho^2$, including the normal dilation and every mixed term. If $G_{-\Delta}$ is the reflected construction’s whole-space fundamental solution, the original fundamental solution is $$G_{p_{\mathrm{orig}}}(x) =\sigma J_F^{-1}G_{-\Delta}(F^{-1}x).$$ To verify the factor, change variables $x=F(w,\rho)$ in a test pairing: the Jacobian $J_F$ cancels $J_F^{-1}$, the two factors of $\sigma$ multiply to one, and $-\Delta G_{-\Delta}=\delta_0$. On a transformed local patch, the original right or left inverse is $f\mapsto [T_{-\Delta}(\sigma f\circ F)]\circ F^{-1}$, with the same support restrictions transported by $F$. All constants in its Sobolev bounds retain their dependence on this fixed map. This conjugation carries every original coefficient through the Laplacian calculation.

Fix a bounded ball $X$ centered on the plane and invariant under reflection. For $f\in L^2(X_+)$, let $Tf$ equal $f$ on $X_+$, its negative reflection on $X_-$, and zero elsewhere. Choose an exact fundamental solution $G$ of $-\Delta$ that is even in $t$. Such a solution is obtained by averaging the solution of Section 2 of Local inverses and distance-weighted elliptic estimates with its reflected distribution. Convolution with $Tf$, restricted to $X_+$, defines $$E_0f=(G*Tf)|_{X_+}. \tag{B13}$$ The local $L^2$ convolution bounds of Section 2 of Local inverses and distance-weighted elliptic estimates give $$\sum_{|\alpha|\leq2}\|D^\alpha E_0f\|_{L^2(X_+)}\leq C_X\|f\|_2, \quad p(D)E_0f=f,\quad\gamma E_0f=0\text{ on }X_0. \tag{B14}$$ For the trace, convolution is odd because $G$ is even and $Tf$ is odd; it is locally $H^2$, so the two one-sided traces agree and oddness forces them to vanish. The convolution bounds are local; a global $L^2$ inverse of the homogeneous Laplacian has not been asserted.

There is also the distinct left identity $$E_0p(D)u_+=u_+,\qquad u\in C_c^\infty(X),\quad u|_{X_0}=0. \tag{B15}$$ To verify it, reflect $u_+$ oddly. The function is continuous at the plane. Its first normal derivative is the even reflection of $\partial_tu_+$, so it too is continuous there. Thus the second normal derivative has no interface delta. Tangential differentiation does not differentiate the sign of $t$, and tangential derivatives of the zero trace vanish. Therefore $p(D)Tu_+=T(p(D)u_+)$; the compared Laplacian has no mixed normal-tangential monomials, and the full original operator is recovered by the proved conjugation. Compact support permits transferring $p(D)$ in the convolution, yielding $G*p(D)Tu_+=(p(D)G)*Tu_+=Tu_+$. This proves (B15).

The same reasoning proves a useful density fact. If $v\in H^2(\mathbb R^n_+)$ has zero trace and compact support up to the plane, its odd extension is $H^2$. The only possible interface term would be the jump of a first derivative. The normal derivative is even, and the tangential derivatives have zero trace by Section 2. Convolving this extension with an even smooth mollifier produces smooth odd functions converging in $H^2$; a symmetric cutoff keeps their supports in any prescribed slightly larger neighborhood. Consequently (B15) and its norm estimate extend to such $H^2$ functions whenever their support lies in the fixed patch.

5. Continuous principal coefficients and a local inverse

Let $$P(x,D)=\sum_{|\alpha|\leq2}a_\alpha(x)D^\alpha \tag{B16}$$ near zero. Suppose the principal coefficients are continuous, their frozen quadratic polynomial is real elliptic, and the lower-order coefficients are locally essentially bounded. Then a sufficiently small neighborhood $X$ has a bounded linear map $E:L^2(X_+)\to H^2(X_+)$ with $$PEf=f,\quad\gamma Ef=0\text{ on }X_0,\qquad EPu_+=u_+\quad(u\in C_c^\infty(X),\ u|_{X_0}=0). \tag{B17}$$ The original continuous-coefficient situation is included. Products in $PEf$ are $L^2$ products, since $Ef\in H^2$.

Proof. Make the boundary-preserving normalization above and work on a fixed unit patch in the rescaled variable $y=x/r$. Multiplying the equation by $r^2$ gives a frozen principal operator $p(D_y)$ and perturbation $$R_r=\sum_{|\alpha|=2}(a_\alpha(ry)-a_\alpha(0))D_y^\alpha +\sum_{|\alpha|<2}r^{2-|\alpha|}a_\alpha(ry)D_y^\alpha. \tag{B18}$$ By (B14), $\|R_rE_0\|_{L^2\to L^2}\to0$. Choose $r$ making this norm at most $1/2$, put $A_r=R_rE_0$, and set $$E_r=E_0(I+A_r)^{-1},\qquad (I+A_r)^{-1}=\sum_{j=0}^\infty(-A_r)^j. \tag{B19}$$ The series converges in operator norm, gives a two-sided inverse, and has norm at most two. Since $P_rE_0=I+A_r$, the right identity and trace assertion follow. For a compact smooth zero-trace $u$, put $g=p(D)u$. Identity (B15) gives $E_0g=u$, and hence $P_ru=(I+A_r)g$. Applying (B19) yields the left identity. Rescaling back gives (B17) and boundedness of every derivative through order two. Explicitly the scaled bound is $\sum_{j=0}^2r^j\|\nabla^j Ef\|_2\leq Cr^2\|f\|_2$. No smallness of unscaled lower-order coefficients was assumed. $\square$

In particular compact smooth zero-trace functions satisfy $$\|w\|_{H^2(X_+)}\leq C\|Pw\|_{L^2(X_+)}. \tag{B20}$$ The odd-density result after (B15) extends (B20) to compactly supported $H^2$ zero-trace functions in a slightly smaller patch. This extension will be used only after the regularized function has been proved to be $H^2$.

6. Recovering second derivatives at a flat boundary

Assume the principal coefficients in (B16) are Lipschitz, the frozen principal polynomial is real elliptic, and the lower-order coefficients are locally bounded. If $$u\in H^1(X_+),\qquad Pu=f\in L^2(X_+),\qquad\gamma u=0\text{ on }X_0, \tag{B21}$$ then for every $Y\Subset X$, $$u\in H^2(Y_+),\qquad \sum_{|\alpha|=2}\|D^\alpha u\|_{L^2(Y_+)} \leq C_Y\bigl(\|f\|_{L^2(X_+)}+\|u\|_{H^1(X_+)}\bigr). \tag{B22}$$ Here $Y$ may meet the boundary plane. Merely knowing the interior theorem would not establish this assertion.

Proof. Choose $\chi\in C_c^\infty(X)$ equal to one near $\overline Y$, put $v=\chi u$, and extend tangentially by zero beyond the patch. The full product rule gives $$Pv=\chi f+ \sum_{|\alpha|\leq2}\sum_{0<\beta\leq\alpha} \binom\alpha\beta a_\alpha(D^\beta\chi)D^{\alpha-\beta}u =:g\in L^2. \tag{BO1}$$ Every derivative on $u$ in the displayed sum has order at most one. Its norm is bounded by $C(\|f\|_2+\|u\|_{H^1})$.

Keep $a=a_{2e_t}$, the original coefficient of $D_t^2$, and shrink the patch so that $|a|\geq a_*>0$. For each other original multi-index put $b_\alpha=a^{-1}a_\alpha$. These are comparison coefficient functions: the equation and estimate will continue to use the original $P$. The principal $b_\alpha$ are Lipschitz; the lower ones are bounded. All bounds retain their dependence on $a_*^{-1}$, $\|a\|_\infty$, its Lipschitz bound and the original coefficients.

Here is the weak multiplication identity needed before second derivatives are known. For $h\in L^2$ and Lipschitz $c,d$, define $cD_jh=D_j(ch)-(D_jc)h$. The weak product rule, tested against compactly supported $H^1$ functions, gives $$d(cD_jh)=D_j(dch)-\big((D_jd)c+d(D_jc)\big)h =(dc)D_jh. \tag{BO2}$$ Indeed multiplication by a Lipschitz function is bounded on the test space $H^1_0$, so it acts by duality on $H^{-1}$; applying the product rule twice gives the displayed terms in their actual order. For $d=a^{-1}$, $c=a$, differentiating $a^{-1}a=1$ makes the entire derivative bracket zero. Thus applying multiplication by $a^{-1}$ to the original weak equation is an exact identity in $H^{-1}$; no second derivative has been assumed.

Let $J_\varepsilon$ be convolution in $z$ alone with a compactly supported smooth approximate identity, and put $v_\varepsilon=J_\varepsilon v$. First derivatives converge in $L^2$. Every second derivative with a tangential factor already belongs to $L^2$, since that factor may differentiate the smooth convolution kernel and the remaining derivative falls on $v\in H^1$. Tangential convolution commutes with $D_t^2$ distributionally. Consequently (BO2) and the complete original equation yield $$\begin{split} aD_t^2v_\varepsilon &=aJ_\varepsilon(a^{-1}g) -a\sum_{\alpha\ne2e_t}J_\varepsilon(b_\alpha D^\alpha v),\\ Pv_\varepsilon &=aJ_\varepsilon(a^{-1}g) +a\sum_{\alpha\ne2e_t} \big(b_\alpha D^\alpha J_\varepsilon v -J_\varepsilon(b_\alpha D^\alpha v)\big). \end{split} \tag{BO3}$$ All sums in this proof range over the original $|\alpha|\leq2$. For a principal $\alpha\ne2e_t$, write $D^\alpha=D_jD_k$ with $j$ tangential. Apply Section 7 of Local inverses and distance-weighted elliptic estimates to $D_kv$ at fixed $t$. It gives $$b_\alpha D^\alpha J_\varepsilon v -J_\varepsilon(b_\alpha D^\alpha v) \longrightarrow0\quad\hbox{in }L^2. \tag{B23}$$ The Lipschitz and convolution bounds are uniform in $t$. More explicitly the commutator is bounded by a fixed constant times $\|D_kv(\cdot,t)\|_{L^2_z}$; for almost every $t$ it tends to zero. The square of this dominating function is integrable, so dominated convergence proves the displayed full-space convergence. For a lower-order term put $h=D^\alpha v\in L^2$. The exact difference $b_\alpha J_\varepsilon h-J_\varepsilon(b_\alpha h) =b_\alpha(J_\varepsilon h-h)+(b_\alpha h-J_\varepsilon(b_\alpha h))$ tends to zero in $L^2$ using only boundedness of $b_\alpha$. Define $$T_\varepsilon g=aJ_\varepsilon(a^{-1}g),\qquad \|T_\varepsilon g-g\|_2 \leq\|a\|_\infty \|J_\varepsilon(a^{-1}g)-a^{-1}g\|_2\longrightarrow0. \tag{BO4}$$ Thus the actual original operator, with its full $D_t^2$ coefficient, satisfies $$Pv_\varepsilon=T_\varepsilon g+o_{L^2}(1). \tag{B24}$$ This identity and the already known nonnormal second derivatives express $aD_t^2v_\varepsilon$ as an $L^2$ function. Multiplication by the retained $a^{-1}$ proves $D_t^2v_\varepsilon\in L^2$, and hence $v_\varepsilon\in H^2$, before (B20) is invoked. Its trace is zero because tangential convolution commutes with the continuous trace; its support lies in a fixed compact subpatch for small $\varepsilon$.

Apply the original estimate (B20) to $v_\varepsilon-v_\delta$, using the odd-extension density after (B15). Formula (B24), with (BO4) and every commutator retained, makes the right side tend to zero. Completeness gives an $H^2$ limit, and the $H^1$ convergence identifies it with $v$. Testing against compact smooth functions identifies all second-derivative limits with its distributional derivatives. The same estimate gives (B22), with constants containing the coefficient and reciprocal bounds already stated. For $n=1$, there is no tangential convolution: the original equation is $aD_t^2v=g-\sum_{\alpha\ne2e_t}a_\alpha D^\alpha v$, whose right side is $L^2$. Its actual nonzero coefficient gives $D_t^2v\in L^2$, and (B20) applies. $\square$

7. Smooth bootstrap and a single inverse on all integer scales

Suppose the coefficients are smooth on a neighborhood of the closed flat patch and retain the ellipticity condition above. For every integer $r\geq0$, a zero-trace solution initially in $H^1$, with $Pu\in H^r$, satisfies on a smaller patch $$\|u\|_{H^{r+2}(Y_+)}\leq C_{r,Y}\bigl(\|Pu\|_{H^r(X_+)}+\|u\|_{H^1(X_+)}\bigr). \tag{B25}$$ The nested patches used in the proof may depend on $r$; the solution does not.

Proof. The case $r=0$ is (B22). Inductively suppose $u\in H^{r+1}$ on the required local patches. For a tangential $\beta$ with $|\beta|=r$, the original differentiated equation is $$PD_z^\beta u =D_z^\beta f- \sum_{|\alpha|\leq2}\sum_{0<\gamma\leq\beta} \binom\beta\gamma(D_z^\gamma a_\alpha) D^\alpha D_z^{\beta-\gamma}u. \tag{BO5}$$ The right side is $L^2$: at least one derivative in each commutator term falls on the original coefficient, leaving at most $r+1$ derivatives on $u$. The function $D_z^\beta u$ belongs to $H^1$, and its trace is zero by (B4) and approximation. Applying (B22) on nested patches gives all derivatives of total order $r+2$ having at most two normal factors.

Retain $a=a_{2e_t}$ and the complete original equation in the form $$aD_t^2u=f-\sum_{\substack{|\alpha|\leq2\\\alpha\ne2e_t}} a_\alpha D^\alpha u. \tag{B26}$$ For a target derivative with $q\geq3$ normal factors, write it as $D^{\beta+2e_t}u$, where $|\beta|=r$ and $\beta_t=q-2$. The full differentiated identity is $$\begin{split} aD^{\beta+2e_t}u ={}&D^\beta f -\sum_{\substack{|\alpha|\leq2\\\alpha\ne2e_t}} \sum_{\gamma\leq\beta} \binom\beta\gamma(D^\gamma a_\alpha) D^{\beta-\gamma+\alpha}u\\ &-\sum_{0<\gamma\leq\beta} \binom\beta\gamma(D^\gamma a) D^{\beta-\gamma+2e_t}u. \end{split} \tag{BO6}$$ In the first sum the terms with $\gamma=0$ have at most $q-1$ normal factors. Every other term on $u$, including every differentiated original normal coefficient in the last sum, has total order at most $r+1$. Increasing induction on $q$, beginning with the tangentially obtained orders, therefore makes the entire right side $L^2$. The weak identity (BO2), or its smooth coefficient version, allows multiplication by the actual $a^{-1}$, which supplies precisely the target derivative and its norm bound. None of the derivatives of $a$ in (BO6) has been removed. Cutoffs between $Y$ and $X$, the full product formula (BO1), and preceding-order estimates control all localization terms. There are finitely many multi-indices at each order, proving (B25) with its stated finite constant. $\square$

There is, in addition, one local inverse acting on the whole half-space, whose higher-regularity property holds for all such integers. Choose a patch $X_1$ on which (B17) is constructed, a smaller $X$, and smooth cutoffs $\eta_0,\eta_1$ with compact support in $X_1$, where $\eta_0=1$ near $\overline X$ and $\eta_1=1$ near $\operatorname{supp}\eta_0$. With $E_1$ the inverse on $(X_1)_+$, define $$\mathcal E f=\bigl[\eta_0 E_1(\eta_1f|_{(X_1)_+})\bigr]^0_{ \text{outside }X_1\text{ within the half-space}}. \tag{B27}$$ The cutoff vanishes near the artificial boundary, so the indicated extension is bounded into $H^2(\mathbb R^n_+)$. On $X_+$, $P\mathcal Ef=f$ and its trace is zero. If $u\in C_c^\infty(X)$ has zero trace, then $\eta_1Pu=Pu$, so (B17) gives $\mathcal EPu=u$ on $X_+$. Finally (B25), applied on a neighborhood of $\operatorname{supp}\eta_0$, gives $$\mathcal E:H^r(\mathbb R^n_+)\longrightarrow H^{r+2}(\mathbb R^n_+) \quad\text{bounded for every integer }r\geq0. \tag{B28}$$ The operator in (B27) is fixed before $r$ is chosen. Global half-space boundedness and the local inverse identities are separate statements.

8. Global regularity of the energy solution

Under the assumptions of Section 3, suppose now that $\partial\Omega$ is $C^{1,1}$ and the coefficients $a_{jk}$ are Lipschitz on $\overline\Omega$. Then the solution of (B10) satisfies $$u\in H^2(\Omega),\qquad \|u\|_{H^2(\Omega)}\leq C\|f\|_2. \tag{B29}$$ If the coefficients, boundary, and forcing are smooth up to the boundary, then $u\in C^\infty(\overline\Omega)$.

Proof. On an interior patch, expansion of divergence form gives $Pu=\sum a_{jk}D_jD_ku+\sum(D_ja_{jk})D_ku$. The second sum is $L^2$, since its coefficients are bounded almost everywhere. The interior theorem in Section 8 of Local inverses and distance-weighted elliptic estimates applies to the principal part with $p=2,m=2$, giving a local $H^2$ bound in terms of $\|f\|_2+\|u\|_{H^1}$.

In a boundary patch use a $C^{1,1}$ flattening $x=F(y)$. Weak $H^1$ chain rules were proved in Section 2. The transformed principal coefficient matrix is obtained by multiplying $A$ on its two sides by the inverse derivative of $F$; it is real positive definite, uniformly so on a smaller compact patch. Its entries are Lipschitz: $A\circ F$ is Lipschitz, and the first derivative of the $C^{1,1}$ diffeomorphism and its inverse are Lipschitz on that patch. The additional first-order coefficients involve first weak derivatives of $A$ and weak second derivatives of $F$, and are bounded almost everywhere. One can first transform the divergence-form weak identity by change of variables and then expand it; this justifies the computation for $u\in H^1$ without assuming second derivatives. The Jacobian and its inverse are bounded. The trace is zero by the characterization after (B6). Thus (B22), including its bounded-lower-coefficient case, gives the desired local estimate. Here are the weak chart details needed at this lower boundary regularity. Locally the boundary is a graph $x_n=h(x')$ with $h\in C^{1,1}$; take $$F(y',t)=(y',t+h(y')),\qquad F^{-1}(x)=(x',x_n-h(x')),\qquad \det DF=1. \tag{BR1}$$ On each fixed compact patch $DF,DF^{-1}$ are bounded and Lipschitz, and every weak second derivative of either map is bounded. A general fixed rigid change before this graph map retains these properties and its actual Jacobian. The $H^1$ composition result in Section 2 applies, since these are $C^1$ diffeomorphisms. For a smooth function $v$, its first composed derivatives are products of $Dv\circ F$ with entries of $DF$. Applying the Lipschitz weak product rule to these products gives the full formula, almost everywhere, $$\partial_i\partial_j(v\circ F) =\sum_{a,b}(\partial_a\partial_bv)\circ F\, \partial_iF_a\,\partial_jF_b +\sum_a(\partial_av)\circ F\,\partial_i\partial_jF_a. \tag{BR2}$$ Every term is in $L^2$, with its norm bounded by a fixed multiple of the $H^2$ norm of $v$ on the corresponding patch. Approximate a localized half-space $H^2$ function by the extension and convolution in Section 2. The zeroth, first and second expressions in (BR2) are Cauchy in $L^2$; testing against compactly supported smooth functions identifies their limits with the weak derivatives of the composed function. This proves the bounded $H^2$ composition map by $F^{-1}$ as well. The input energy solution needs only the already proved $H^1$ composition until the flattened boundary theorem supplies its $H^2$ regularity.

For clarity, the original divergence-form weak identity transforms with the full coefficient $$\mathcal A(y)=|\det DF(y)|\,DF(y)^{-1} A(F(y))\,DF(y)^{-T}. \tag{BR3}$$ Change of variables in the two first-derivative integrals proves this identity for $H^1$ inputs and tests. Divide by the positive Jacobian to write the transformed distributional equation on coordinate volume. Expanding the divergence uses weak first derivatives of the Lipschitz entries of $\mathcal A$; they are bounded. The transformed principal matrix is $DF^{-1}A(F)DF^{-T}$, and is uniformly positive on the retained compact patch. All remaining terms involve bounded coefficient derivatives, the retained Jacobian derivatives and first derivatives of the solution. The right side is $f\circ F$, with its actual volume norm. This proves the nondivergence-form interface to (B22) under $C^{1,1}$, without assuming classical second chart derivatives. After (B22), (BR2) transfers the resulting $H^2$ function back. Thus (B29) includes $C^2$ boundaries and the larger $C^{1,1}$ class with the same original operator and energy domain.

A finite cover of the compact closure by these boundary patches and interior patches, with cutoffs, yields $\|u\|_{H^2}\leq C(\|f\|_2+\|u\|_{H^1})$. Bound (B10) removes the final term and proves (B29). For smooth data use (B25) in every boundary chart and its interior version, or the same differentiated-equation argument without a boundary. This gives every integer Sobolev order. To conclude classical smoothness, extend each localized function by the integer extension after (B4); for $k>j+n/2$, Cauchy-Schwarz makes $\xi^\alpha\widehat u\in L^1$, $|\alpha|\leq j$. Fourier inversion and dominated convergence give continuous derivatives through order $j$. Since $j$ is arbitrary, smoothness holds up to the boundary. $\square$

A boundary with a bounded second weak derivative and no classical second derivative. In a two-dimensional chart take $h(z)=z|z|$, with the domain above its graph. Then $h'(z)=2|z|$ is Lipschitz, and $h''=2\operatorname{sgn}z$ almost everywhere; the second derivative has no value at zero. The graph flattening (BR1) has determinant one and the transformed full negative Laplacian is $$-\partial_z^2+4|z|\partial_z\partial_t -(1+4z^2)\partial_t^2 +2\operatorname{sgn}z\,\partial_t =-\operatorname{div}\! \left(\begin{pmatrix}1&-2|z|\\-2|z|&1+4z^2\end{pmatrix} \nabla\right). \tag{BR7}$$ Both formulas act weakly. The matrix determinant is one and its quadratic form is $(\xi_z-2|z|\xi_t)^2+\xi_t^2$; on compact patches it is uniformly positive. Its entries are Lipschitz, and its displayed lower-order coefficient is bounded. Thus this graph satisfies the exact boundary regularity interface although it is not $C^2$. It is a local chart example; the global theorem still requires the full bounded domain and its complete boundary to be $C^{1,1}$.

The two graph pieces and the transported grid are numerical samples of the exact formulas (BR1) and (BR7). The weak derivative has values $2$ and $-2$ on the two open sides; its value at zero does not affect the operator. The $H^2$ receiving map is proved in (BR2)–(BR3) and (B29).

9. Complex quadratic symbols and the low-frequency repair

The construction below supplies the complex-coefficient extension of the local Dirichlet inverse and, with smooth coefficients, its simultaneous integer Sobolev improvements. An auxiliary boundary at a fixed positive height removes the low-frequency obstruction. It disappears from the final local assertions.

9.0. Unordered roots, simple-root coordinates and the tangential sphere

We prove the finite-dimensional root and sphere facts before using them in the normal-mode construction. The full factorization theorem, with the original leading coefficient and every multiplicity, is proved in Sections9.1–9.4 of Polynomial and contour tools. Closed bounded finite-coordinate sets are compact by Sections12.6–12.9 of Metric foundations. The actual inverse and implicit maps, including their full remainders and all higher derivatives, are proved in Sections16.4–16.5 of Geometric microlocal calculus. Below we identify the exact polynomial and real derivative matrices to which those proofs apply. No half-plane root count is used to prove its own continuity prerequisite.

The original quadratic, including a repeated root

Let $$Q(z)=az^2+bz+c,\qquad a\ne0,\qquad \mathcal D=b^2-4ac,\qquad d^2=\mathcal D. \tag{RC1}$$ The complex-root theorem supplies a square root of the actual number $\mathcal D$. When $\mathcal D=0$, its square root is zero. If $\mathcal D\ne0$, its two square roots are $d,-d$: any other square root $e$ obeys $(e-d)(e+d)=0$, so one factor is zero. Put $$z_1=\frac{-b+d}{2a},\qquad z_2=\frac{-b-d}{2a},\qquad Q(z)=a(z-z_1)(z-z_2). \tag{RC2}$$ To prove the last identity, add and multiply the first two expressions: their sum is $-b/a$, and their product is $(b^2-d^2)/(4a^2)=c/a$. Expanding the full last product therefore gives exactly $az^2+bz+c$. The original leading factor $a$, both occurrences of a repeated root, and the entire discriminant remain. Changing $d$ to $-d$ exchanges the two entries and changes no unordered pair. When $\mathcal D=0$, both entries are the original double root $-b/(2a)$.

For two unordered root lists $R=(z_1,z_2)$, $R'=(w_1,w_2)$, define their distance by $$d_2(R,R')= \min\{\max(|z_1-w_1|,|z_2-w_2|), \max(|z_1-w_2|,|z_2-w_1|)\}. \tag{RC3}$$ Permuting either list exchanges the two candidates, so this is well-defined for unordered pairs with multiplicity. It is symmetric and nonnegative, and it vanishes exactly when the lists agree after a permutation. The triangle inequality follows by composing minimizing permutations for two consecutive comparisons and applying the ordinary complex triangle inequality to each matched entry, then taking the maximum. Thus RC3 is a metric on these unordered pairs. It does not choose a continuous square-root sign or discard repeated roots.

Suppose the original coefficient triples $(a_k,b_k,c_k)$ tend to $(a,b,c)$, with $a\ne0$. For all sufficiently large $k$, $|a_k|\geq|a|/2$, and $|b_k|+|c_k|$ has a finite bound $M$. Every root $z$ of the actual $Q_k$ with $|z|\geq1$ satisfies $$|a_k|\,|z|^2\leq |b_k|\,|z|+|c_k| \leq (|b_k|+|c_k|)|z|,\qquad |z|\leq\frac{2M}{|a|}. \tag{RC4}$$ Consequently all roots lie in the fixed disk of radius $1+2M/|a|$. List the two roots in nondecreasing order of their original real coordinates and then their imaginary coordinates, retaining repeats. This selects representatives of the unordered pairs; no continuity of the ordering is asserted. Finite-coordinate compactness gives a subsequence on which both entries converge, say to $u,v$. Passing to the limit in the complete factorization, or in each of its three coefficients, gives $$az^2+bz+c=a(z-u)(z-v). \tag{RC5}$$ Uniqueness of multiplicities in the full factorization theorem says that $(u,v)$ is exactly the original unordered root pair of $Q$. If convergence in RC3 failed, a subsequence would have distance at least some $\varepsilon>0$ from that pair. Applying the preceding compactness argument to this subsequence gives a further subsequence whose distance tends to zero, a contradiction. This also proves epsilon-delta continuity: its failure would provide coefficient triples within $1/k$ of the original triple whose root distances stay at least one fixed positive epsilon, contradicting the just-proved sequence result. Thus continuity holds at every original coefficient triple with nonzero leading coefficient, including the discriminant-zero locus. An ordered continuous choice was neither assumed nor needed.

The following finite-degree extension is separate from the quadratic entry, and uses the same full factorization proof. For original polynomials $$P_k(z)=\sum_{j=0}^m a_{j,k}z^j,\quad a_{j,k}\longrightarrow a_j,\quad a_m\ne0,\qquad P(z)=\sum_{j=0}^m a_jz^j, \tag{RC6}$$ keep the degree $m$, leading coefficient and all $m$ root occurrences. For $|z|\geq1$, the original root equation gives $|a_{m,k}||z|^m\leq\sum_{j=0}^{m-1}|a_{j,k}||z|^j \leq |z|^{m-1}\sum_{j=0}^{m-1}|a_{j,k}|$. The nonvanishing leading coefficient and bounded complete lower-coefficient sum give a fixed disk containing every root. Any sequence of full root lists has a coordinatewise convergent subsequence. In the limit the original identity is $P(z)=a_m\prod_{j=1}^m(z-u_j)$. Full factorization uniqueness identifies every limit occurrence with its correct multiplicity. The same contradiction proves convergence in $$d_m(R,R')=\min_{\sigma\in\mathfrak S_m} \max_{1\leq j\leq m}|z_j-w_{\sigma(j)}|. \tag{RC7}$$ The proof that this is an unordered-multiset metric is exactly the composition-of-permutations argument, with all $m$ matched terms retained. A nonzero constant has the empty root list and distance zero; the zero polynomial and a vanishing degree-$m$ leading coefficient are outside this theorem. No polynomial is replaced by a monic one.