Gradient method for nondensely defined closed unbounded linear operators

Complex Powers of Nondensely Defined Operators

The power (−A)b, b ∈ C is defined for a closed linear operator A whose resolvent is polynomially bounded on the region which is, in general, strictly contained in an acute angle. It is proved that all structural properties of complex powers of densely defined operators with polynomially bounded resolvent remain true in the newly arisen situation. The fractional powers are considered as generato...

Dynamical Systems Method (dsm) for Unbounded Operators

Let L be an unbounded linear operator in a real Hilbert space H, a generator of a C0 semigroup, and let g : H → H be a C2 loc nonlinear map. The DSM (dynamical systems method) for solving equation F (v) := Lv+ g(v) = 0 consists of solving the Cauchy problem u̇ = Φ(t, u), u(0) = u0, where Φ is a suitable operator, and proving that i) ∃u(t) ∀t > 0, ii) ∃u(∞), and iii) F (u(∞)) = 0. Conditions on L...

Existence Results for Semilinear Perturbed Functional Differential Equations with Nondensely Defined Operators

A level-set method for computing the eigenvalues of elliptic operators defined on closed surfaces

We reduce the calculation of the eigenvalues of an elliptic operator defined on a closed and bounded surface in R to the solution of an elliptic eigenvalue problem in divergence form in R via separation of variables and estimates from semi-classical analysis. By representing the surface implicitly, we solve the latter problem using standard finite element methods on a regular mesh. In an append...

Relatively Closed Operators and Linear Evolution Equations

In this paper it is shown how the vector-valued Laplace transform can be used to obtain existence and uniqueness results for exponentially bounded solutions of abstract linear evolution equations in Banach spaces. The method allows to consider problems for which the characteristic operator is, like many sums, products or limits of closed operators, not necessarily closed or closable.