Search

Abstract: In this paper we investigate vector-valued parabolic initial boundary value problems of relaxation type. Typical examples for such boundary conditions are dynamic boundary conditions or linearized free boundary value problems like in the Stefan problem. We present a complete -theory for such problems which is based on maximal regularity of certain model problems. [Copyright &y& Elsevier]

Abstract: We explore the connection between square-integrable solutions for real-values of the spectral parameter λ and the continuous spectrum of self-adjoint ordinary differential operators with arbitrary deficiency index d. We show that if, for all λ in an open interval I, there are d of linearly independent square-integrable solutions, then for every extension of the point spectrum is nowhere dense in I, and there is a self-adjoint extension of which has no continuous spectrum in I. This analysis is based on our construction of limit-point (LP) and limit-circle (LC) solutions obtained recently in an earlier paper. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the study of Lyapunov-type inequalities () for linear partial differential equations. More precisely, we treat the case of Neumann boundary conditions on bounded and regular domains in . It is proved that the relation between the quantities p and plays a crucial role. This fact shows a deep difference with respect to the ordinary case. The linear study is combined with Schauder fixed point theorem to provide new conditions about the existence and uniqueness of solutions for resonant nonlinear problems. [Copyright &y& Elsevier]

Abstract: This paper proves that any initial condition in the energy space for the plate equation with square root damping on a smooth bounded domain, with hinged boundary conditions , can be steered to zero by a square integrable input function u supported in arbitrarily small time interval and subdomain. As T tends to zero, for initial states with unit energy norm, the norm of this u grows at most like for any real and some . Indeed, this fast controllability cost estimate is proved for more general linear elastic systems with structural damping and non-structural controls satisfying a spectral observability condition. Moreover, under some geometric optics condition on the subdomain allowing to apply the control transmutation method, this estimate is improved into and the dependence of on the subdomain is made explicit. These results are analogous to the optimal ones known for the heat flow. [Copyright &y& Elsevier]

Abstract: A connection between the semigroup of the Cauchy process killed upon exiting a domain D and a mixed boundary value problem for the Laplacian in one dimension higher known as the mixed Steklov problem, was established in [R. Bañuelos, T. Kulczycki, The Cauchy process and the Steklov problem, J. Funct. Anal. 211 (2004) 355–423]. From this, a variational characterization for the eigenvalues , , of the Cauchy process in D was obtained. In this paper we obtain a variational characterization of the difference between and . We study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound estimates for where is the eigenvalue corresponding to the “first” antisymmetric eigenfunction for D. The proof is based on a variational characterization of and on a weighted Poincaré-type inequality. The Poincaré inequality is valid for all α symmetric stable processes, , and any other process obtained from Brownian motion by subordination. We also prove upper bound estimates for the spectral gap in bounded convex domains. [Copyright &y& Elsevier]