Weak solvability of the positive scalar conductivity equation

In dimension one, endpoint evaluation is bounded on $H^1$ by the fundamental theorem of calculus and Cauchy--Schwarz. Any endpoint data have a smooth collar lift. Its kernel is $H_0^1$: for a zero-endpoint function, $\mathrm{\mid }u(t){\mathrm{\mid }}^2\le t{\int }_0^t\mathrm{\mid }{u}^{\mathrm{\prime }}(r){\mathrm{\mid }}^{2}dr$ makes the derivative error of a cutoff on $[0,2\delta ]$ tend to zero in $L^2$ as $\delta \to 0$; cut off at both endpoints and mollify. Thus the trace, lift and kernel conclusions of [F5], [F9] and [F10] hold also in this dimension, without invoking their higher-dimensional Euclidean suppliers. There is a constant $C_P$ such that $\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2(M)}\le C_P\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2(M)}$ for $w\in H_0^1(M)$. Otherwise choose $w_j\in H_0^1(M)$ with $\mathrm{\parallel }{w}_j{\mathrm{\parallel }}_{L^2}=1$ and $\mathrm{\parallel }\mathrm{\nabla }{w}_j{\mathrm{\parallel }}_{L^2}\to 0$. A finite chart partition and [F2] give an $L^2$-convergent subsequence. On compact $M$, the fixed finite chartwise $H^1$ norm is equivalent to $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{L^2}+\mathrm{\parallel }{\mathrm{\nabla }}_g\cdot {\mathrm{\parallel }}_{L^2}$ by bounded metric coefficients, so this subsequence is Cauchy in $H^1$ and converges to some $w\in H_0^1(M)$ with $\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2}=1$ and weak gradient zero. Local convolution shows that a function with zero weak gradient is constant on each connected coordinate ball; overlapping balls and connectedness make $w$ constant on $M^{\circ }$. Its trace is zero because $w\in H_0^1(M)$, while [F9] makes the trace of a nonzero constant equal that same nonzero boundary value. This contradiction proves the inequality.

Compactness of $M$ and positivity of $\gamma$ give ${\gamma }_{\ast }:={\min }_M\gamma >0$. Thus $a_{\gamma }(w,z)={\int }_M\gamma g(\mathrm{\nabla }w,\mathrm{\nabla }\bar{z})d{V}_g$ is bounded on $H_0^1(M)$ and $a_{\gamma }(w,w)\ge c\mathrm{\parallel }w{\mathrm{\parallel }}_{H^1(M)}^2$ by step 1.1 and ${\gamma }_{\ast }$. By [F3], for any bounded conjugate-linear functional $G$ on $H_0^1(M)$ there is a unique $w\in H_0^1(M)$ with $a_{\gamma }(w,z)=G(z)$ for all $z$.

Flatten each boundary chart, apply the local half-space trace lift, and patch the finitely many chart lifts with a smooth partition. The resulting $Ef\in H^1(M)$ has trace $f$ and norm bounded by $C\mathrm{\parallel }f{\mathrm{\parallel }}_{H^{1\mathrm{/}2}(\mathrm{\partial }M)}$ by [F5]. For $h\in L^2(M)$ take $G(z)=-{\int }_{M}h\bar{z}d{V}_g-a_{\gamma }(Ef,z)$ in step 2.1 and set $v=Ef+w$. Then $v$ has trace $f$ and satisfies the weak equation. If $v_1,v_2$ are two such solutions, their difference has zero global trace, hence lies in $H_0^1(M)$ by [F10]. Its energy is zero; since $\gamma \ge {\gamma }_{\ast }>0$, its gradient vanishes, and step 1.1 gives $v_1=v_2$.

Suppose $h$ and $f$ are smooth. By [F8], extend $f$ smoothly from the boundary into a collar and multiply by a cutoff to obtain a smooth extension $Ef$ on $M$. Then $w=v-Ef$ has zero trace, so $w\in H_0^1(M)$ by [F10], and solves ${\mathrm{div}}_g(\gamma {\mathrm{\nabla }}_{g}w)=h-{\mathrm{div}}_g(\gamma {\mathrm{\nabla }}_{g}Ef)$, whose right-hand side is smooth. For dimension one, the coordinate equation $(a{w}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}=H$ with smooth $a>0$ and smooth $H$ first gives $a{w}^{\mathrm{\prime }}\in H^1$ and then, by iteration, $w\in H^m$ for every $m$; the one-dimensional fundamental theorem gives smoothness up to the endpoints. For dimension at least two, in each boundary chart, flatten the boundary and write the principal matrix as $\mathrm{\mid }g{\mathrm{\mid }}^{1\mathrm{/}2}\gamma g^{ij}$; it is smooth and uniformly elliptic because $g$ is positive definite and $\gamma$ has a positive minimum. For the base case, localize $w$ by a cutoff; its equation has an $L^2$ right-hand side since $w\in H^1$, so [F6] gives local $H^2$ regularity. If $w\in H^{m+1}$, each cutoff commutator contains derivatives of $w$ of order at most $m+1$ with smooth bounded coefficients, hence lies in $H^m$; the localized right-hand side is therefore in $H^m$, and [F6] gives $H^{m+2}$ regularity. Induction over $m$ and a finite chart cover give $w\in H^m(M)$ for every integer $m$. By [F7], $w$ and hence $v=Ef+w$ are smooth up to the boundary.