Том 233, № 4 (2018)

We consider some fully nonlinear degenerate elliptic operators and we investigate the validity of certain properties related to the maximum principle. In particular, we establish the equivalence between the sign propagation property and the strict positivity of a suitably defined generalized principal eigenvalue. Furthermore, we show that even in the degenerate case considered in the present paper, the well-known condition introduced by Keller–Osserman on the zero-order term is necessary and sufficient for the existence of entire weak subsolutions.

We describe the current state of the theory of equations with m-Hessian stationary and evolution operators. It is quite important that new algebraic and geometric notions appear in this theory. In the present work, a list of those notions is provided. Among them, the notion of m-positivity of matrices is quite important; we provide a proof of an analog of Sylvester’s criterion for such matrices. From this criterion, we easily obtain necessary and sufficient conditions for existence of classical solutions of the first initial boundary-value problem for m-Hessian evolution equations. The asymptotic behavior of m-Hessian evolutions in a semibounded cylinder is considered as well.

In the space ℍ = (L₂[0, π])², we study the Dirac operator \( {\mathrm{\mathcal{L}}}_{P,U} \) generated by the differential expression ℓ_P(y) = By′ + Py, where

\( B=\left(\begin{array}{cc}-i& 0\\ {}0& i\end{array}\right),\kern0.5em P(x)=\left(\begin{array}{cc}{p}_1(x)& {p}_2(x)\\ {}{p}_3(x)& {p}_4(x)\end{array}\right),\kern0.5em \mathbf{y}(x)=\left(\begin{array}{c}{y}_1(x)\\ {}{y}_2(x)\end{array}\right), \)

and the regular boundary conditions

\( U\left(\mathbf{y}\right)=\left(\begin{array}{cc}{u}_{11}& {u}_{12}\\ {}{u}_{21}& {u}_{22}\end{array}\right)\left(\begin{array}{c}{y}_1(0)\\ {}{y}_2(0)\end{array}\right)+\left(\begin{array}{cc}{u}_{13}& {u}_{14}\\ {}{u}_{23}& {u}_{24}\end{array}\right)\left(\begin{array}{c}{y}_1\left(\uppi \right)\\ {}{y}_2\left(\uppi \right)\end{array}\right)=0. \)

The elements of the matrix P are assumed to be complex-valued functions summable over [0, π]. We show that the spectrum of the operator \( {\mathrm{\mathcal{L}}}_{P,U} \) is discrete and consists of eigenvalues {λ_n}_{n ∈ ℤ} such that \( {\uplambda}_n={\uplambda}_n^0+o(1) \) as |n| → ∞, where \( {\left\{{\uplambda}_n^0\right\}}_{n\in \mathrm{\mathbb{Z}}} \) is the spectrum of the operator \( {\mathrm{\mathcal{L}}}_{0,U} \) with zero potential and the same boundary conditions. If the boundary conditions are strongly regular, then the spectrum of the operator \( {\mathrm{\mathcal{L}}}_{P,U} \) is asymptotically simple. We show that the system of eigenfunctions and associate functions of the operator \( {\mathrm{\mathcal{L}}}_{P,U} \) forms a Riesz base in the space ℍ provided that the eigenfunctions are normed. If the boundary conditions are regular, but not strongly regular, then all eigenvalues of the operator \( {\mathrm{\mathcal{L}}}_{0,U} \) are double, all eigenvalues of the operator \( {\mathrm{\mathcal{L}}}_{P,U} \) are asymptotically double, and the system formed by the corresponding two-dimensional root subspaces of the operator \( {\mathrm{\mathcal{L}}}_{P,U} \) is a Riesz base of subspaces (Riesz base with brackets) in the space ℍ.

We study the well-posedness of initial-value problems for abstract integrodifferential equations with unbounded operator coefficients in Hilbert spaces and provide a spectral analysis of operator functions that are symbols of the specified equations. These equations represent an abstract form of linear partial integrodifferential equations arising in viscoelasticity theory and other important applications. For the said integrodifferential equations, we obtain well-posedness results in weighted Sobolev spaces of vector functions defined on the positive semiaxis and valued in a Hilbert space. For the symbols of the said equations, we find the localization and the structure of the spectrum.