2008/07/31 by Lucian Maticiuc, Etienne Pardoux, Aurel Răşcanu +1 · 1 citation
Computer Science · Mathematics · #Contact Mechanics and Variational Inequalities #Function (biology) #Nonlinear Partial Differential Equations #Operator (biology) #Optimization and Variational Analysis #Parabolic partial differential equation #Subderivative #Uniqueness #Variational inequality #Viscosity #Viscosity solution #Weak solution #math.AP #math.DS
paper · pdf · doi:10.3150/09-bej204
published as Bernoulli, vol. 16, no. 1, 258 - 273 (2010) · Published in at http://dx.doi.org/10.3150/09-BEJ204 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)
openalex publication_date 2010/02/01 · arxiv created 2010/02/23 · arxiv updated 2015/10/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
In this paper, we first define the notion of viscosity solution for the following system of partial differential equations involving a subdifferential operator: \cases \dfrac∂ u∂ t(t,x )+Ltu(t,x )+f (t,x,u (t,x )) ∈∂φ (u (t,x)), t∈[ 0,T) ,x∈ℝd,\cr u( T,x ) =h(x), x∈ℝd, where ∂φ is the subdifferential operator of the proper convex lower semicontinuous function φ:ℝk→ (-∞,+∞] and Lt is a second differential operator given by Ltvi(x)=(1)/(2)Tr% [σ(t,x)σ∗(t,x)D2vi(x) ]+ ⟨ b(t,x),∇ vi(x) ⟩, i∈1,k. We prove the uniqueness of the viscosity solution and then, via a stochastic approach, prove the existence of a viscosity solution u: [ 0,T ] ×ℝd→ℝk of the above parabolic variational inequality.