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Trace Operator
In mathematics, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space. This is particularly important for the study of partial differential equations with prescribed boundary conditions ( boundary value problems), where weak solutions may not be regular enough to satisfy the boundary conditions in the classical sense of functions. Motivation On a bounded, smooth domain \Omega \subset \mathbb R^n, consider the problem of solving Poisson's equation with inhomogeneous Dirichlet boundary conditions: :\begin -\Delta u &= f &\quad&\text \Omega,\\ u &= g &&\text \partial \Omega \end with given functions f and g with regularity discussed in the application section below. The weak solution u \in H^1(\Omega) of this equation must satisfy :\int_\Omega \nabla u \cdot \nabla \varphi \,\mathrm dx = \int_\Omega f \varphi \,\mathrm dx for all \varphi \in H^1_0(\Omega). The H^1(\Omega)-regularity of u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Operator Illustration
Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' (magazine), British hip-hop magazine * ''Trace'' (manhwa), a Korean internet cartoon * ''Trace'' (novel), a novel by Patricia Cornwell * ''The Trace'' (film), a 1994 Turkish film * ''The Trace'' (video game), 2015 video game * ''Sama'' (film), alternate title ''The Trace'', a 1988 Tunisian film * Trace, a fictional character in the game ''Metroid Prime Hunters'' * Trace, the protagonist of ''Axiom Verge'' * Trace, another name for Portgas D. Ace, a fictional character in the manga ''One Piece'' * TRACE, the main brand for a number of music channels such as Trace Urban Language * Trace (deconstruction), a concept in Derridian deconstruction * Trace (linguistics), a syntactic placeholder resulting from a transformation * TRACE (psycho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Test Function
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or where appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function. A function f is normally thought of as on the in the function domain by "sending" a point x in its domain to the point f(x). Instead of acting on points, distribution theory reinterpre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
