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p Spaces
Published in Kenneth Kuttler, Modern Analysis, 2017
Definition 12.18 Let U = {x ∈ ℝn: ‖x‖ < 1}. A sequence {ψm}⊆Cc∞(U) is called a mollifier (sometimes an approximate identity) if
n
Published in H-H Dai, P L Sachdev, Recent Advances in Differential Equations, 2020
Proof If w is a smooth function, then it follows immediately. In general, we can construct a sequence of smooth functions Wh using mollifier which converge to w as h � 0. It can be seen that Wh satisfies the same inequality as w. Hence, w must blow up in finite time. ? Proof of Theorem 1.3. First we consider the case ? n(p � 1). It follows from (2.7') that
General Introduction
Published in Didier Felbacq, Guy Bouchitté, Metamaterials Modeling and Design, 2017
Didier Felbacq, André Nicolet, Frédéric Zolla
Another step could have undermined the range of validity of Maxwell equations: the transition from the microscopic to the macroscopic level. For instance, the movement of individual molecules in a gas can be described satisfactorily using Newton's laws of mechanics, but their collective behavior at the macroscopic level is more efficiently expressed in terms of the laws of thermodynamics. Maxwell equations work well in a void for an electromagnetic field possibly interacting with given charges (the charges create electromagnetic fields, and the electromagnetic fields act on the charges via the Lorentz force). Consider now the interaction of an incident electromagnetic field with matter from a macroscopic point of view: Even if the considered piece of matter is globally neutral, it is made of a huge number of charged electrons and protons interacting with the electromagnetic field at the microscopic level. How can we describe the resulting electromagnetic field since it will suffer hectic fluctuations in space and time due, for instance, to the influence of microscopic charges shaken by thermal agitation? Fortunately, macroscopic equations can be found for an averaged (in space and time) electromagnetic field that takes into account only the large-scale behavior of the field. This smoothed electromagnetic field is named macroscopic electromagnetic field and it is described by Maxwell equations! The reason is that the averaging process is a low-pass filtering performed by convoluting the field with a smooth bounded support function, the mollifier, and this operation commutes with the partial derivations with respect to space coordinates and time. Therefore, the equations for the macroscopic electromagnetic field retain their initial form. Of course, something has to change to take into account the interaction with matter: The free space constitutive relations involving the permittivity ε0 and the permeability μ0 are replaced by ad hoc constitutive relations involving for instance homogeneous linear isotropic permittivity, permeability, and conductivity or possibly much more unwieldy models. The generic term for this process is homogenization.
Mandel's problem as a benchmark for two-dimensional nonlinear poroelasticity
Published in Applicable Analysis, 2022
C. J. van Duijn, A. Mikelić, T. Wick
Using the weak topology of W, difficulties arise with the continuity of the coefficients and K depending on div . To remedy this, we introduce, as in [13], a Friedrichs mollifier , where ε is a small positive parameter (see e.g. [28] page 203), and replace div in the nonlinearities by the convolution div . Using this substitution one can treat nonlinear coefficients containing div as lower-order terms in the equations. This allows us to use the theory of pseudo-monotone operators.
Real-time reconstruction of moving point/dipole wave sources from boundary measurements
Published in Inverse Problems in Science and Engineering, 2020
Throughout this section, we assume that and . Then, from Proposition 2.1 and since is a three-dimensional smooth manifold, observation data φ is in . For the reconstruction of moving point sources, we choose the following five sequences of functions in : where , , and denotes the standard mollifier function with the support (e.g. Appendix 3 in [36]). We note that sequences and have already applied to the reconstruction of fixed point sources [19]. Supplemental sequences and are used to treat the effect of moving velocities of sources.
A non-singular continuum theory of point defects using gradient elasticity of bi-Helmholtz type
Published in Philosophical Magazine, 2019
If we use the Green tensor (48), then the solution of Equation (85) is the convolution of the (negative) Green tensor with the right hand side of Equation (85)where the ficticious body force density reads asEquation (86) is the generalised Volterra formula for an arbitrary eigendistortion. The gradient of Equation (86) gives the displacement gradientOn the other hand, using the regularisation function or mollifier of bi-Helmholtz type, , the solutions of Equations (82) and (83) can be written as convolution of the classical singular solution with the mollifier