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2. 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.

3. Green's function for the three-variable Laplace equation

   en.wikipedia.org/wiki/Green's_function_for_the...

   Using the Green's function for the three-variable Laplace operator, one can integrate the Poisson equation in order to determine the potential function. Green's functions can be expanded in terms of the basis elements (harmonic functions) which are determined using the separable coordinate systems for the linear partial differential equation ...

4. Infinity Laplacian - Wikipedia

   en.wikipedia.org/wiki/Infinity_Laplacian

   In this equation, we used sup and inf instead of max and min because the graph (,) does not have to be locally finite (i.e., to have finite degrees): a key example is when () is the set of points in a domain in , and (,) if their Euclidean distance is at most . The importance of this example lies in the following.

5. Elliptic operator - Wikipedia

   en.wikipedia.org/wiki/Elliptic_operator

   A solution to Laplace's equation defined on an annulus.The Laplace operator is the most famous example of an elliptic operator.. In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator.

6. Cylindrical harmonics - Wikipedia

   en.wikipedia.org/wiki/Cylindrical_harmonics

   The cylindrical harmonics for (k,n) are now the product of these solutions and the general solution to Laplace's equation is given by a linear combination of these solutions: (,,) = | | (,) (,) where the () are constants with respect to the cylindrical coordinates and the limits of the summation and integration are determined by the boundary ...

7. Laplacian vector field - Wikipedia

   en.wikipedia.org/wiki/Laplacian_vector_field

   However, the converse is not true; not every vector field that satisfies Laplace's equation is a Laplacian vector field, which can be a point of confusion. For example, the vector field v = ( x y , y z , z x ) {\displaystyle {\bf {v}}=(xy,yz,zx)} satisfies Laplace's equation, but it has both nonzero divergence and nonzero curl and is not a ...

8. Laplace operators in differential geometry - Wikipedia

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

   In dimension n ≥ 3, the conformal Laplacian, denoted L, acts on a smooth function u by = +, where Δ is the Laplace-Beltrami operator (of negative spectrum), and R is the scalar curvature. This operator often makes an appearance when studying how the scalar curvature behaves under a conformal change of a Riemannian metric.

9. Well-posed problem - Wikipedia

   en.wikipedia.org/wiki/Well-posed_problem

   Examples of archetypal well-posed problems include the Dirichlet problem for Laplace's equation, and the heat equation with specified initial conditions. These might be regarded as 'natural' problems in that there are physical processes modelled by these problems. Problems that are not well-posed in the sense above are termed ill-posed. A ...