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RAR2: Recent advances in regularity theory for PDEs Location: M628 Session Chair: Simon Nowak Session Chair: Marvin Weidner | |
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Nonlocal gradients in variational problems: Heterogeneous horizons and local boundary conditions 1: KU Eichstätt-Ingolstadt, Germany; 2: UCLouvain, Belgium Building on recent advances in nonlocal hyperelasticity, we discuss a class of variational problems involving integral functionals with nonlocal gradients. Specific to our set-up is a space-dependent interaction range that vanishes at the boundary of the reference domain. This ensures that the operator depends only on values within the domain and localizes to the classical gradient at the boundary, which allows for a seamless integration of nonlocal modeling with local boundary values. Our main contribution is a comprehensive study of the associated Sobolev spaces, including embedding results, the analysis of a trace operator and the proof of a Poincaré inequality. A central aspect of our technical approach lies in exploiting connections with pseudodifferential operator theory. As an application, we establish the existence of minimizers for functionals with quasiconvex or polyconvex integrands depending on heterogeneous nonlocal gradients, subject to local Dirichlet-, Neumann- or mixed-type boundary conditions. | |
