Weak well-posedness of the anisotropic conductivity equation

The space $H_0^1(\mathrm{\Omega })$ is a closed linear subspace of the complete space $H^1(\mathrm{\Omega })$ by [F1] and [F3]. Since $\mathrm{\Omega }$ is bounded, it lies in a slab $a<x\cdot e<b$ for some unit vector $e$ and finite $a<b$; applying [F2] with $p=2$ gives $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2}\le C(2)(b-a)\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2}$ for every $v\in H_0^1(\mathrm{\Omega })$. Thus $\mathrm{\parallel }v{\mathrm{\parallel }}_H:=\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2}$ is a norm equivalent to the inherited $H^1$ norm, and $H_0^1(\mathrm{\Omega })$ is a real Hilbert space for $(v,w{)}_H={\int }_{\mathrm{\Omega }}\mathrm{\nabla }v\cdot \mathrm{\nabla }wdx$.

Fix any representative $U\in H^1(\mathrm{\Omega })$ of $f$; the existence argument below is independent of which representative is fixed. On $H_0^1(\mathrm{\Omega })$ define $a_{\gamma }(z,v)={\int }_{\mathrm{\Omega }}\gamma \mathrm{\nabla }z\cdot \mathrm{\nabla }vdx$ and $F_U(v)=-{\int }_{\mathrm{\Omega }}\gamma \mathrm{\nabla }U\cdot \mathrm{\nabla }vdx$. By [F6], $a_{\gamma }$ is bounded by $C\mathrm{\parallel }z{\mathrm{\parallel }}_H\mathrm{\parallel }v{\mathrm{\parallel }}_H$, coercive by $c\mathrm{\parallel }z{\mathrm{\parallel }}_H^2$, and $F_U$ is bounded by $C\mathrm{\parallel }\mathrm{\nabla }U{\mathrm{\parallel }}_{L^2}\mathrm{\parallel }v{\mathrm{\parallel }}_H$. Countable Choice follows from [F5], so [F4] gives a unique $z\in H_0^1(\mathrm{\Omega })$ with $a_{\gamma }(z,v)=F_U(v)$ for every $v\in H_0^1(\mathrm{\Omega })$. Then $u_{\gamma ,f}:=U+z$ has quotient class $f$ and satisfies the stated weak equation.

If $u_1,u_2$ are two weak solutions with boundary class $f$, then $w=u_1-u_2\in H_0^1(\mathrm{\Omega })$ and ${\int }_{\mathrm{\Omega }}\gamma \mathrm{\nabla }w\cdot \mathrm{\nabla }vdx=0$ for every $v\in H_0^1(\mathrm{\Omega })$. Taking $v=w$ and using [F6] gives $c\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2}^2\le 0$, so $\mathrm{\parallel }w{\mathrm{\parallel }}_H=0$ and hence $w=0$ by the norm property in step 1.1.

Let $h\in {\mathcal{B}}_{\mathrm{\Omega }}$ and let $V,V^{\mathrm{\prime }}$ be any two representatives of $h$. Their difference lies in $H_0^1(\mathrm{\Omega })$ by [F1], so the weak equation for $u_{\gamma ,f}$ gives ${\int }_{\mathrm{\Omega }}\gamma \mathrm{\nabla }{u}_{\gamma ,f}\cdot \mathrm{\nabla }(V-V^{\mathrm{\prime }})dx=0$. Therefore the energy pairing is independent of the representative of $h$.

The weak solution minimizes the quadratic energy in its boundary class: if $W$ is any representative of $f$, then $W-u_{\gamma ,f}\in H_0^1(\mathrm{\Omega })$, and expanding the symmetric quadratic form with the weak equation gives ${\int }_{\mathrm{\Omega }}(\gamma \mathrm{\nabla }W)\cdot \mathrm{\nabla }Wdx={\int }_{\mathrm{\Omega }}(\gamma \mathrm{\nabla }{u}_{\gamma ,f})\cdot \mathrm{\nabla }{u}_{\gamma ,f}dx+{\int }_{\mathrm{\Omega }}(\gamma \mathrm{\nabla }(W-u_{\gamma ,f}))\cdot \mathrm{\nabla }(W-u_{\gamma ,f})dx$. Using [F6] and taking the infimum over representatives yields $\mathrm{\parallel }\mathrm{\nabla }{u}_{\gamma ,f}{\mathrm{\parallel }}_{L^2}\le \sqrt{C\mathrm{/}c}\mathrm{\parallel }f{\mathrm{\parallel }}_{{\mathcal{B}}_{\mathrm{\Omega }}}$. The same infimum argument gives representatives $V$ of $h$ with $\mathrm{\parallel }V{\mathrm{\parallel }}_{H^1}\le \mathrm{\parallel }h{\mathrm{\parallel }}_{{\mathcal{B}}_{\mathrm{\Omega }}}+\epsilon$ for every $\epsilon >0$. Cauchy–Schwarz and the upper bound in [F6] now give $\mathrm{\mid }⟨{\mathrm{\Lambda }}_{\gamma }f,h⟩\mathrm{\mid }\le C\sqrt{C\mathrm{/}c}\mathrm{\parallel }f{\mathrm{\parallel }}_{{\mathcal{B}}_{\mathrm{\Omega }}}(\mathrm{\parallel }h{\mathrm{\parallel }}_{{\mathcal{B}}_{\mathrm{\Omega }}}+\epsilon )$; letting $\epsilon \downarrow 0$ proves boundedness. To prove bilinearity, for $f_1,f_2\in {\mathcal{B}}_{\mathrm{\Omega }}$ and $a,b\in \mathbb{R}$, the function $a{u}_{\gamma ,f_1}+b{u}_{\gamma ,f_2}$ has boundary class $a{f}_1+b{f}_2$ and satisfies the weak equation by linearity of the quotient map and of the integral. Uniqueness from step 3.1 therefore gives $u_{\gamma ,a{f}_1+b{f}_2}=a{u}_{\gamma ,f_1}+b{u}_{\gamma ,f_2}$. For $h_1,h_2\in {\mathcal{B}}_{\mathrm{\Omega }}$, representatives $V_{h_1},V_{h_2}$ give $a{V}_{h_1}+b{V}_{h_2}$ as a representative of $a{h}_1+b{h}_2$; representative independence from step 3.2 and linearity of the integral show that the energy pairing is linear in its second argument as well. Thus the energy form is bilinear, and the bound just proved makes it a bounded bilinear form on ${\mathcal{B}}_{\mathrm{\Omega }}\times {\mathcal{B}}_{\mathrm{\Omega }}$.