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Pseudo-differential operators
Published in Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2018
The Sobolev spaces and Fourier transform provide the basic tools we shall need in our study of elliptic partial differential operators. Let () x=(x1,…,xm)∈Rm.
Hilbert Spaces
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
Trace Property. An abstraction of the idea of boundary values of functions from a Hilbert space, exemplified in the Trace Theorem for Sobolev spaces, is embodied in the concept of spaces with a trace property. A Hilbert space V is said to have the trace property if the following conditions hold:V is continuously embedded in a larger Hilbert space HAV↪H $$ AV \hookrightarrow H $$ Note that this in particular implies that the topology of H, when restricted to V, is weaker than the original topology of V.There exists a linear and continuous (trace) operator γ $ \gamma $ that maps Vonto another (boundary) Hilbert space ∂V $ \partial V $ such that the kernel of γ $ \gamma $ , denoted V0 $ V_{0} $ , is everywhere dense in H: AV0¯=H,V0=kerγ=N(γ) $$ A\overline{V_{0}} = H,\quad V_{0} = \text{ ker} \gamma = \mathcal N (\gamma ) $$ It follows that the original space V is dense in H as well.
Singular (p,q)-equations with competing perturbations
Published in Applicable Analysis, 2022
Nikolaos S. Papageorgiou, Chao Zhang
The two main spaces that we will use in the study of problem (1) are the Sobolev space and the Banach space . By we denote the norm of . On account of the Poincaré inequality, we have The Banach space is an ordered with positive (order) cone This cone has a nonempty interior given by with being the normal derivative, where is outward unit normal on .
Acoustic wave propagation in anisotropic media with applications to piezoelectric materials
Published in Applicable Analysis, 2022
denote our underlying Hilbert space. We endow with the weight The assumptions (H1)–(H4) ensure that the matrix M (and its inverse) are positive definite. The inner product on will be where is the usual inner product. Now we need the following function space: where the divergence is understood in the weak sense. The norm on this space is given by We also define A useful characterization of this space is the following (see, e.g. [7]): where denotes the standard Sobolev space of functions having weak derivative in . Let denote the space of functions on Ω that have vanishing trace. Then we can define the following unbounded operator: The following result is key for well-posedness of solutions:
Existence results for mixed hemivariational-like inequalities involving set-valued maps
Published in Optimization, 2021
The space is a Banach space w.r.t. the so-called Luxemburg norm and it is reflexive if and only if If are two N-functions then the embedding is continuous if and only if Also note that if and and the Hölder inequality for Orlicz spaces reads as follows The Orlicz–Sobolev space is defined as the space of all such that and it is a Banach space w.r.t. the norm Moreover, if Φ, then is a reflexive and separable space.