On systems of parabolic variational inequalities with multivalued terms

In this paper we present an analytical framework for the following system of multivalued parabolic variational inequalities in a cylindrical domain $$Q=\varOmega \times (0,\tau )$$ Q = Ω × ( 0 , τ ) : For $$k=1,\dots , m$$ k = 1 , ⋯ , m , find $$u_k\in K_k$$ u k ∈ K k and $$\eta _k\in L^{p'_k}(Q)$$ η k ∈ L p k ′ ( Q ) such that $$\begin{aligned}&u_k(\cdot ,0)=0\ \text{ in } \varOmega ,\ \ \eta _k(x,t)\in f_k(x,t,u_1(x,t), \dots , u_m(x,t)), \\&\langle u_{kt}+A_k u_k, v_k-u_k\rangle +\int _Q \eta _k\, (v_k-u_k)\,dxdt\ge 0,\ \ \forall \ v_k\in K_k, \end{aligned}$$ u k ( · , 0 ) = 0 in Ω , η k ( x , t ) ∈ f k ( x , t , u 1 ( x , t ) , ⋯ , u m ( x , t ) ) , ⟨ u kt + A k u k , v k - u k ⟩ + ∫ Q η k ( v k - u k ) d x d t ≥ 0 , ∀ v k ∈ K k , where $$K_k $$ K k is a closed and convex subset of $$L^{p_k}(0,\tau ;W_0^{1,p_k}(\varOmega ))$$ L p k ( 0 , τ ; W 0 1 , p k ( Ω ) ) , $$A_k$$ A k is a time-dependent quasilinear elliptic operator, and $$f_k:Q\times \mathbb {R}^m\rightarrow 2^{\mathbb {R}}$$ f k : Q × R m → 2 R is an upper semicontinuous multivalued function with respect to $$s\in {\mathbb R}^m$$ s ∈ R m . We provide an existence theory for the above system under certain coercivity assumptions. In the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence and enclosure results. As an application, a multivalued parabolic obstacle system is treated. Moreover, under a lattice condition on the constraints $$K_k$$ K k , systems of evolutionary variational-hemivariational inequalities are shown to be a subclass of the above system of multivalued parabolic variational inequalities.

Finite Rank Operators in Certain Algebras

Let Alg(ℒ) be the algebra of all bounded linear operators on a normed linear space X leaving invariant each member of the complete lattice of closed subspaces L. We discuss when the subalgebra of finite rank operators in Alg(ℒ) is non-zero, and give an example which shows this subalgebra may be zero even for finite lattices. We then give a necessary and sufficient lattice condition for decomposing a finite rank operator F into a sum of a rank one operator and an operator whose range is smaller than that of F, each of which lies in Alg(ℒ). This unifies results of Erdos, Longstaff, Lambrou, and Spanoudakis. Finally, we use the existence of finite rank operators in certain algebras to characterize the spectra of Riesz operators (generalizing results of Ringrose and Clauss) and compute the Jacobson radical for closed algebras of Riesz operators and Alg(ℒ) for various types of lattices.