|
Elliptic Operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions. Elliptic operators are typical of potential theory, and they appear frequently in electrostatics and continuum mechanics. Elliptic regularity implies that their solutions tend to be smooth functions (if the coefficients in the operator are smooth). Steady-state solutions to hyperbolic and parabolic equations generally solve elliptic equations. Definitions Let L be a linear differential operator of order ''m'' on a domain \Omega in R''n'' given by Lu = \sum_ a_\alpha(x)\partial^\alpha u where \alpha = (\alpha_1, \dots, \alpha_n) denotes a multi-index, and \partial^\alpha u = \partial^_1 \cdots \partial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ice Sheet Dynamics
In glaciology, an ice sheet, also known as a continental glacier, is a mass of glacial ice that covers surrounding terrain and is greater than . The only current ice sheets are the Antarctic ice sheet and the Greenland ice sheet. Ice sheets are bigger than ice shelves or alpine glaciers. Masses of ice covering less than 50,000 km2 are termed an ice cap. An ice cap will typically feed a series of glaciers around its periphery. Although the surface is cold, the base of an ice sheet is generally warmer due to geothermal heat. In places, melting occurs and the melt-water lubricates the ice sheet so that it flows more rapidly. This process produces fast-flowing channels in the ice sheet — these are ice streams. Even stable ice sheets are continually in motion as the ice gradually flows outward from the central plateau, which is the tallest point of the ice sheet, and towards the margins. The ice sheet slope is low around the plateau but increases steeply at the margins ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not address boundary conditions). In terms of the Dirac delta "function" , a fundamental solution is a solution of the inhomogeneous equation Here is ''a priori'' only assumed to be a distribution. This concept has long been utilized for the Laplacian in two and three dimensions. It was investigated for all dimensions for the Laplacian by Marcel Riesz. The existence of a fundamental solution for any operator with constant coefficients — the most important case, directly linked to the possibility of using convolution to solve an arbitrary right hand side — was shown by Bernard Malgrange and Leon Ehrenpreis, and a proof is available in Joel Smoller (1994). In the context of functional analysis, fundamental solutions are usually ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypoelliptic Operator
In the theory of partial differential equations, a partial differential operator P defined on an open subset :U \subset^n is called hypoelliptic if for every distribution u defined on an open subset V \subset U such that Pu is C^\infty (smooth), u must also be C^\infty. If this assertion holds with C^\infty replaced by real-analytic, then P is said to be ''analytically hypoelliptic''. Every elliptic operator with C^\infty coefficients is hypoelliptic. In particular, the Laplacian is an example of a hypoelliptic operator (the Laplacian is also analytically hypoelliptic). In addition, the operator for the heat equation (P(u)=u_t - k\,\Delta u\,) :P= \partial_t - k\,\Delta_x\, (where k>0) is hypoelliptic but not elliptic. However, the operator for the wave equation The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or elec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sobolev Inequality
In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under slightly stronger conditions some Sobolev spaces are compactly embedded in others. They are named after Sergei Lvovich Sobolev. Sobolev embedding theorem Let denote the Sobolev space consisting of all real-valued functions on whose weak derivatives up to order are functions in . Here is a non-negative integer and . The first part of the Sobolev embedding theorem states that if , and are two real numbers such that :\frac-\frac = \frac -\frac, (given n, p, k and \ell this is satisfied for some q \in [1, \infty) provided (k- \ell) p n, the embedding criterion will hold with r=0 and some positive value of \alpha. That is, for a function f on \mathbb R^n, if f has k derivatives in L^p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact closure in Y). Such an operator is necessarily a bounded operator, and so continuous. Some authors require that X,Y are Banach, but the definition can be extended to more general spaces. Any bounded operator ''T'' that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalization of the class of finite-rank operators in an infinite-dimensional setting. When ''Y'' is a Hilbert space, it is true that any compact operator is a limit of finite-rank operators, so that the class of compact operators can be defined alternatively as the closure of the set of finite-rank operators in the norm topology. Whether this was true in general for Banach spaces (the approximation property) was an unsolved que ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sobolev Space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense. Motivation In this section and throughout the article \Omega is an open subset of \R^n. There are man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Solution
In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for different classes of equations. One of the most important is based on the notion of distributions. Avoiding the language of distributions, one starts with a differential equation and rewrites it in such a way that no derivatives of the solution of the equation show up (the new form is called the weak formulation, and the solutions to it are called weak solutions). Somewhat surprisingly, a differential equation may have solutions that are not differentiable, and the weak formulation allows one to find such solutions. Weak solutions are important because many differential equations encountered in modelling real-world phenomena do not admit of suff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Alternative
In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue. Linear algebra If ''V'' is an ''n''-dimensional vector space and T:V\to V is a linear transformation, then exactly one of the following holds: #For each vector ''v'' in ''V'' there is a vector ''u'' in ''V'' so that T(u) = v. In other words: ''T'' is surjective (and so also bijective, since ''V'' is finite-dimensional). #\dim(\ker(T)) > 0. A more elementary formulation, in terms of matrices, is as follows. Given an ''m''×''n'' matrix ''A'' and a ''m''×1 column vector b, exactly one of the following must hold: #''Either:'' ''A'' x = b has a solution x #''Or:'' ''A''T y = 0 has a so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gårding's Inequality
In mathematics, Gårding's inequality is a result that gives a lower bound for the bilinear form induced by a real linear elliptic partial differential operator. The inequality is named after Lars Gårding. Statement of the inequality Let \Omega be a bounded, open domain in n-dimensional Euclidean space and let H^k(\Omega) denote the Sobolev space of k-times weakly differentiable functions u\colon\Omega\rightarrow\mathbb with weak derivatives in L^2(\Omega). Assume that \Omega satisfies the k-extension property, i.e., that there exists a bounded linear operator E\colon H^k(\Omega)\rightarrow H^k(\mathbb^n) such that Eu\vert_\Omega=u for all u\in H^k(\Omega). Let ''L'' be a linear partial differential operator of even order ''2k'', written in divergence form :(L u)(x) = \sum_ (-1)^ \mathrm^ \left( A_ (x) \mathrm^ u(x) \right), and suppose that ''L'' is uniformly elliptic, i.e., there exists a constant ''θ'' > 0 such that :\sum_ \xi^ A_ (x) \xi^ > \theta , \xi , ^ \mbox x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
