David Natroshvili, Maia Svanadze
abstract:
We investigate mixed boundary value problems (BVP) of the linear theory of
viscoelasticity for isotropic and homogeneous Kelvin-Voigt materials with voids
when on one part of the boundary of the body under consideration the Dirichlet
type condition is given and on the remaining part of the boundary the Neumann
type condition is prescribed. Using the potential method and the theory of
pseudodifferential equations we prove the existence and uniqueness of solutions
in the appropriate Sobolev-Slobodetskii, Bessel potential, and Besov spaces.
Using the embedding theorems, we establish almost optimal regularity results for
solutions to the mixed BVPs near the collision curves where different types of
boundary conditions collide. In particular, we prove that the solutions belong
to the space of H\"older continuous functions in the closed region occupied by
the viscoelastic body. An efficient algebraic algorithm is described for finding
the Hölder smoothness exponents which, in
turn, efficiently determined the corresponding stress singularity exponents near
the collision curves. It is shown that these exponents depend essentially on the
material parameters.
Mathematics Subject Classification: 31B10, 35C15, 45A05, 45F15, 47G40, 74D05, 74G25
Key words and phrases: Viscoelasticity, potential theory, pseudodifferential equations, mixed boundary value problem, regularity of solutions