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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.
Solutions of the Laplace equation, i.e. functions whose Laplacian is identically zero, thus represent possible equilibrium densities under diffusion. The Laplace operator itself has a physical interpretation for non-equilibrium diffusion as the extent to which a point represents a source or sink of chemical concentration, in a sense made ...
In general solutions of equations involving the p-Laplacian do not have second order derivatives in classical sense, thus solutions to these equations have to be understood as weak solutions. For example, we say that a function u belonging to the Sobolev space W 1 , p ( Ω ) {\displaystyle W^{1,p}(\Omega )} is a weak solution of
In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace.It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location parameter) spliced together along the abscissa, although the term is also sometimes used to refer to ...
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.
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.
When viewed as a distribution the second partial derivative's values can be changed at an arbitrary set of points as long as this has Lebesgue measure 0. Since in the example the Hessian is symmetric everywhere except (0, 0), there is no contradiction with the fact that the Hessian, viewed as a Schwartz distribution, is symmetric.
The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.