Abstract
Abstract Given an open set Ω ⊂ Rm and n > 1, we introduce the new spaces GBnV(Ω) of Generalized functions of bounded higher variation and GSBnV(Ω) of Generalized special functions of bounded higher variation that generalize, respectively, the space BnV introduced by Jerrard and Soner in [43] and the corresponding SBnV space studied by De Lellis in [24]. In this class of spaces, which allow as in [43] the description of singularities of codimension n, the distributional jacobian Ju need not have finite mass: roughly speaking, finiteness of mass is not required for the (m−n)-dimensional part of Ju, but only finiteness of size. In the space GSBnV we are able to provide compactness of sublevel sets and lower semicontinuity of Mumford-Shah type functionals, in the same spirit of the codimension 1 theory [5,6].
Highlights
The space BV of functions of bounded variation, consisting of real-valued functions u defined in a domain of Rm whose distributional derivative Du is a finite Radon measure, may contain discontinuous functions and, precisely for this reason, can be used to model a variety of phenomena, while on the PDE side it plays an important role in the theory of conservation laws [20, 14]
De Giorgi and the first author introduced the distinguished subspace SBV of special functions of bounded variation, whose distributional derivative consists of an absolutely continuous part and a singular part concentrated on a (m − 1)-dimensional set, called discontinuity set Su
For this reason, having in mind application to the Ginzburg-Landau theory Jerrard and Soner introduced in [43] the space BnV of functions of bounded higher variation, where n stands for the codimension: roughly speaking it consists of Sobolev maps u : Ω → Rn whose distributional Jacobian Ju is representable by a vector-valued measure: in this case the natural vector space is the space Λm−nRm of (m − n)-vectors
Summary
The space BV of functions of bounded variation, consisting of real-valued functions u defined in a domain of Rm whose distributional derivative Du is a finite Radon measure, may contain discontinuous functions and, precisely for this reason, can be used to model a variety of phenomena, while on the PDE side it plays an important role in the theory of conservation laws [20, 14]. Only the H m−1-dimensional measure of Suh does not provide a control on the width of the jump This difficulty leads [22] to the space GSBV of generalized special functions of bounded variation, i.e. the space of all real-valued maps u whose truncates (−N ) ∨ u ∧ N are all SBV. For these reasons, when looking for compactness properties in SBnV , we have been led to define the space GSBnV of generalized special functions of bounded higher variation as the space of functions u such that Ju is representable in the form R + T , with R absolutely continuous with respect to L m and T having finite size in an appropriate sense, made rigorous by the slicing theory of flat currents (in the same vein, one can define GBnV , but our main object of investigation will be GSBnV ).
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
