Potentials, method of

A method for studying boundary value problems in mathematical physics by reducing them to integral equations; this method consists in representing the solutions of these problems in the form of (generalized) potentials.

Let a second-order elliptic partial differential equation be given in $ \mathbf R ^ {n} $, $ n \geq 2 $,

$$ \tag{1 } L u \equiv \sum _ {i , j = 1 } ^ { n } \left ( a _ {ij} \frac{\partial ^ {2} u }{\partial x _ {i} \partial x _ {j} } \right ) + \sum _ {i = 1 } ^ { n } e _ {i} \frac{\partial u }{\partial x _ {i} } + c u = f ( x), $$

with sufficiently smooth coefficients $ a _ {ij} = a _ {ji} = a _ {ij} ( x) $, $ e _ {i} = e _ {i} ( x) $, $ c ( x) \leq 0 $, and right-hand side $ f ( x) $; moreover, let $ c ( x) < - k ^ {2} < 0 $ outside some bounded domain containing in its interior a domain $ D $ with boundary $ S = \partial D $ of class $ C ^ {1} $. Then any solution $ u ( x) $ of class $ C ^ {2} ( D \cup S ) $ of (1) can be represented as the sum of three (generalized) potentials: a volume mass potential (cf. Newton potential)

$$ \tag{2 } \int\limits _ { D } E ( x , y ) \rho ( y) d y , $$

a single-layer potential (cf. Simple-layer potential)

$$ \tag{3 } \int\limits _ { S } E ( x , y ) \sigma ( y) \ d s _ {y} , $$

and a double-layer potential

$$ \tag{4 } \int\limits _ { S } Q _ {y} [ E ( x , y ) ] \mu ( y) \ d s _ {y} , $$

where $ E ( x , y) $ is a principal fundamental solution of $ L $, the symbol $ Q _ {y} $ denotes the operator

$$ Q _ {y} \nu = a \frac{\partial \nu }{\partial N } - b \nu , $$

acting at a point $ y \in S $, $ N $ is a unit co-normal vector at the point $ y \in S $,

$$ a ^ {2} = \sum _ { i= } 1 ^ { n } \left ( \sum _ { j= } 1 ^ { n } a _ {ij} \cos ( \nu , y _ {j} ) \right ) ^ {2} ,\ b = \sum _ { i= } 1 ^ { n } e _ {i} \cos ( \nu , y _ {i} ) , $$

and $ \nu $ is the exterior normal vector to $ S $ at $ y \in S $. The potential densities $ \rho ( y) $, $ \sigma ( y) $ and $ \mu ( y) $ are sufficiently-smooth functions in $ D $ or on $ S $.

All differentiability and boundary properties of harmonic potentials described in the article Potential theory for the case when $ L $ is the Laplace operator are valid for the potentials (2)–(4). On the basis of these properties one can reduce boundary value problems for elliptic equations of type (1) to integral equations in the same way as it has been done in the case of the Dirichlet and Neumann problems for harmonic functions in the article Potential theory.

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