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  2. Laplace operator - Wikipedia

    en.wikipedia.org/wiki/Laplace_operator

    The Laplace operator is a second-order differential operator in the n-dimensional Euclidean space, defined as the divergence of the gradient (). Thus if f {\displaystyle f} is a twice-differentiable real-valued function , then the Laplacian of f {\displaystyle f} is the real-valued function defined by:

  3. Laplace's equation - Wikipedia

    en.wikipedia.org/wiki/Laplace's_equation

    In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties.This is often written as = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.

  4. Laplace operators in differential geometry - Wikipedia

    en.wikipedia.org/wiki/Laplace_operators_in...

    The Hodge Laplacian, also known as the Laplace–de Rham operator, is a differential operator acting on differential forms. (Abstractly, it is a second order operator on each exterior power of the cotangent bundle.) This operator is defined on any manifold equipped with a Riemannian- or pseudo-Riemannian metric.

  5. Elliptic partial differential equation - Wikipedia

    en.wikipedia.org/wiki/Elliptic_partial...

    The simplest example of a second-order linear elliptic PDE is the Laplace equation, in which a i,j is zero if i ≠ j and is one otherwise, and where b i = c = f = 0. The Poisson equation is a slightly more general second-order linear elliptic PDE, in which f is not required to vanish.

  6. Infinity Laplacian - Wikipedia

    en.wikipedia.org/wiki/Infinity_Laplacian

    In mathematics, the infinity Laplace (or -Laplace) operator is a 2nd-order partial differential operator, commonly abbreviated .It is alternately defined, for a function : of the variables = (, …,), by

  7. Elliptic operator - Wikipedia

    en.wikipedia.org/wiki/Elliptic_operator

    This is the most general form of a second-order divergence form linear elliptic differential operator. The Laplace operator is obtained by taking A = I. These operators also occur in electrostatics in polarized media. Example 3

  8. Discrete Laplace operator - Wikipedia

    en.wikipedia.org/wiki/Discrete_Laplace_operator

    Discrete Laplace operator is often used in image processing e.g. in edge detection and motion estimation applications. [4] The discrete Laplacian is defined as the sum of the second derivatives and calculated as sum of differences over the nearest neighbours of the central pixel. Since derivative filters are often sensitive to noise in an image ...

  9. p-Laplacian - Wikipedia

    en.wikipedia.org/wiki/P-Laplacian

    In mathematics, the p-Laplacian, or the p-Laplace operator, is a quasilinear elliptic partial differential operator of 2nd order. It is a nonlinear generalization of the Laplace operator , where p {\displaystyle p} is allowed to range over 1 < p < ∞ {\displaystyle 1<p<\infty } .