Condenser capacity

Let Σ be a closed, smooth, (n − 1)-dimensional hypersurface in n-dimensional Euclidean space ℝⁿ, n ≥ 3; K will denote the n-dimensional compact (i.e., closed and bounded) set of which Σ is the boundary. Let S be another (n − 1)-dimensional hypersurface that encloses Σ: in reference to its origins in electromagnetism, the pair (Σ, S) is known as a condenser. The condenser capacity of Σ relative to S, denoted C(Σ, S) or cap(Σ, S), is given by the surface integral

${\displaystyle C(\Sigma ,S)=-{\frac {1}{(n-2)\sigma _{n}}}\int _{S'}{\frac {\partial u}{\partial \nu }}\,\mathrm {d} \sigma ',}$

where:

• u is the unique harmonic function defined on the region D between Σ and S with the boundary conditions u(x) = 1 on Σ and u(x) = 0 on S;
• S′ is any intermediate surface between Σ and S;
• ν is the outward unit normal field to S′ and

${\displaystyle {\frac {\partial u}{\partial \nu }}(x)=\nabla u(x)\cdot \nu (x)}$

is the normal derivative of u across S′; and

• σ_n = 2π^n⁄2 ⁄ Γ(n ⁄ 2) is the surface area of the unit sphere in ℝⁿ.

C(Σ, S) can be equivalently defined by the volume integral

${\displaystyle C(\Sigma ,S)={\frac {1}{(n-2)\sigma _{n}}}\int _{D}|\nabla u|^{2}\mathrm {d} x.}$

The condenser capacity also has a variational characterization: C(Σ, S) is the infimum of the Dirichlet's energy functional

${\displaystyle I[v]={\frac {1}{(n-2)\sigma _{n}}}\int _{D}|\nabla v|^{2}\mathrm {d} x}$

over all continuously-differentiable functions v on D with v(x) = 1 on Σ and v(x) = 0 on S.

Harmonic/Newtonian capacity

Heuristically, the harmonic capacity of K, the region bounded by Σ, can be found by taking the condenser capacity of Σ with respect to infinity. More precisely, let u be the harmonic function in the complement of K satisfying u = 1 on Σ and u(x) → 0 as x → ∞. Thus u is the Newtonian potential of the simple layer Σ. Then the harmonic capacity (also known as the Newtonian capacity) of K, denoted C(K) or cap(K), is then defined by

${\displaystyle C(K)=\int _{\mathbb {R} ^{n}\setminus K}|\nabla u|^{2}\mathrm {d} x.}$

If S is a rectifiable hypersurface completely enclosing K, then the harmonic capacity can be equivalently rewritten as the integral over S of the outward normal derivative of u:

${\displaystyle C(K)=\int _{S}{\frac {\partial u}{\partial \nu }}\,\mathrm {d} \sigma .}$

The harmonic capacity can also be understood as a limit of the condenser capacity. To wit, let S_r denote the sphere of radius r about the origin in ℝⁿ. Since K is bounded, for sufficiently large r, S_r will enclose K and (Σ, S_r) will form a condenser pair. The harmonic capacity is then the limit as r tends to infinity:

${\displaystyle C(K)=\lim _{r\to \infty }C(\Sigma ,S_{r}).}$

The harmonic capacity is a mathematically abstract version of the electrostatic capacity of the conductor K and is always non-negative and finite: 0 ≤ C(K) < +∞.

Divergence form elliptic operators

Solutions to a uniformly elliptic partial differential equation with divergence form

${\displaystyle \nabla \cdot (A\nabla u)=0}$

are minimizers of the associated energy functional

${\displaystyle I[u]=\int _{D}(\nabla u)^{T}A(\nabla u)\,\mathrm {d} x}$

subject to appropriate boundary conditions.

The capacity of a set E with respect to a domain D containing E is defined as the infimum of the energy over all continuously-differentiable functions v on D with v(x) = 1 on E; and v(x) = 0 on the boundary of D.

The minimum energy is achieved by a function known as the capacitary potential of E with respect to D, and it solves the obstacle problem on D with the obstacle function provided by the indicator function of E. The capacitary potential is alternately characterized as the unique solution of the equation with the appropriate boundary conditions.