In this paper we study the optimal control of system driven by hemivariational inequality of second order. First, we establish the existence of solutions to hemivariational inequality which contains nonlinear pseudomonotone evolution operator. Introducing a control variable in the multivalued term of the generalized subdifferential, we prove the closedness (in suitable topologies) of the graph ...

We study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). In particular, we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a Euclidean space. This inequality controls the kth eigenvalue by the lower eigenvalues, independently of the particular geometry of the domain. Our inequality is shar...

Let (M, g, σ) be a closed Riemannian spin manifold of dimension n ≥ 2. We define the two quantities λ±min(M, [g], σ) := infg∈[g] |λ ± 1 (g)|Vol(M, g) , where λ+1 (g) is the smallest positive eigenvalue of the Dirac operator D and λ−1 (g) is its largest negative eigenvalue. If S is the sphere S with the standard metric, it is known that the two inequalities λ±min(M, [g], σ) ≤ λ + min(S ) hold. I...

In this article we investigate spectral properties of the coupling H +Vλ, where H = −iα · ∇ +mβ is the free Dirac operator in R, m > 0 and Vλ is an electrostatic shell potential (which depends on a parameter λ ∈ R) located on the boundary of a smooth domain in R. Our main result is an isoperimetric-type inequality for the admissible range of λ’s for which the coupling H + Vλ generates pure poin...

If T is a complex symmetric operator on a separable complex Hilbert space H, then the spectrum σ(|T |) of √ T ∗T can be characterized in terms of a certain approximate antilinear eigenvalue problem. This approach leads to a general inequality (applicable to any bounded operator T : H → H), in terms of the spectra of the selfadjoint operators ReT and ImT , restricting the possible location of el...

We study the boundedness of Riesz transforms in L for p > 2 on a doubling metric measure space endowed with a gradient operator and an injective, ω-accretive operator L satisfying Davies-Gaffney estimates. If L is non-negative self-adjoint, we show that under a reverse Hölder inequality, the Riesz transform is always bounded on L for p in some interval [2, 2 + ε), and that L gradient estimates ...

We obtain new proofs with improved constants of the Khintchine-type inequality with matrix coefficients in two cases. The first case is the Pisier and Lust-Piquard noncommutative Khintchine inequality for p = 1 , where we obtain the sharp lower bound of 1 √ 2 in the complex Gaussian case and for the sequence of functions {en}n=1 . The second case is Junge’s recent Khintchine-type inequality for...