Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Laplacian of the indicator
open-in-new
<p>In <a href="/facts/Potential_theory/3sXGIcDA">potential theory</a> (a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>), the Laplacian of the indicator is obtained by letting the <a href="/facts/Laplace_operator/kov9u3PT">Laplace operator</a> work on the <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> of some <a href="/facts/Domain_(mathematics)/gMPmK8Ps">domain</a> <i>D</i>. It is a generalisation of the <a href="/facts/Derivative_(mathematics)/7Ito70Dh">derivative</a> (or "prime function") of the <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a> to <a href="/facts/Higher_dimension/0ktmUyac">higher dimensions</a>; it is non-zero only on the <a href="/facts/Surface_(mathematics)/whF08jvy">surface</a> of <i>D</i>. It can be viewed as a <i>surface delta prime function</i>, the derivative of a <i>surface delta function</i> (a generalization of the Dirac delta). The Laplacian of the indicator is also analogous to the <a href="/facts/Second_derivative/3aA6j4EY">second derivative</a> of the <a href="/facts/Heaviside_step_function/bgsUtxgP">Heaviside step function</a> in one dimension. </p><p>The Laplacian of the indicator can be thought of as having infinitely positive and negative values when evaluated very near the boundary of the domain <i>D</i>. Therefore, it is not strictly a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> but a <a href="/facts/Generalized_function/fCM4VsyT">generalized function</a> or <a href="/facts/Measure_(mathematics)/yCq7zlI4">measure</a>. Similarly to the derivative of the Dirac delta function in one dimension, the Laplacian of the indicator only makes sense as a mathematical object when it appears under an integral sign; i.e. it is a <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">distribution</a> function. Just as in the formulation of distribution theory, it is in practice regarded as a limit of a sequence of <a href="/facts/Smooth_function/mffL4ch2">smooth functions</a>; one may meaningfully take the Laplacian of a <a href="/facts/Bump_function/ExdBtMHN">bump function</a>, which is smooth by definition, and let the bump function approach the indicator in the <a href="/facts/Limit_(mathematics)/j8JBhj4e">limit</a>. </p>