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{{Short description|Method for reconstructing a harmonic function in a domain}}
{{About|the method in mathematics|the hair painting technique|Hair highlighting#Hair painting}} 
In [[potential theory]], a mathematical discipline, '''balayage''' (from French: ''[[wiktionary:fr:balayage|balayage]]'' "scanning, sweeping") is a method devised by [[Henri Poincaré]] for reconstructing an [[harmonic function]] in a domain from its values on the boundary of the domain.<ref>{{SpringerEOM| title=Balayage method | id=Balayage_method | oldid=11819 | first=E.D. | last=Solomentsev }}</ref>

In modern terms, the '''balayage operator''' maps a [[Measure (mathematics)|measure]] ''&mu;'' on a closed domain ''D'' to a measure ''&nu;'' on the boundary ''&part; D'', so that the [[Newtonian potential]]s of ''&mu;'' and ''&nu;'' coincide outside <math>\bar D</math>. The procedure is called balayage since the mass is "swept out" from ''D'' onto the boundary.

For ''x'' in ''D'', the balayage of [[Dirac delta function|''&delta;''<sub>''x''</sub>]] yields the [[harmonic measure]] ''&nu;''<sub>''x''</sub> corresponding to ''x''. Then the value of a harmonic function ''f'' at ''x'' is equal to

:<math> f(x) = \int_{\partial D} f(y) \, d\nu_x(y).</math>

== References ==
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{{DEFAULTSORT:Balayage}}
[[Category:Potential theory]]


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