A weak formulation for singular symmetric differential expressions is presented in spaces of functions which possess minimal differentiability requirements. These spaces are used to characterize the domains of the various operators associated with such expressions. In particular, domains of self-adjoint differential operators are characterized.

We construct a degree for mappings of the form $F+K$ between Banach spaces, where $F$ is $C^1$ Fredholm of index $0$ and $K$ is compact. This degree generalizes both the Leray-Schauder degree when $F=I$ and the degree for $C^1$ Fredholm mappings of index $0$ when $K=0$. To exemplify the use of this degree, we prove the “invariance-of-domain” property when $F+K$ is one-to-one and a generalization of Rabinowitz's global bifurcation theorem for equations $F\left(\lambda ,x\right)+K\left(\lambda ,x\right)=0$.

The paper is devoted to general elliptic problems in the Douglis-Nirenberg sense. We obtain a necessary and sufficient condition of normal solvability in the case of unbounded domains. Along with the ellipticity condition, proper ellipticity and Lopatinsky condition that determine normal solvability of elliptic problems in bounded domains, one more condition formulated in terms of limiting problems should be imposed in the case of unbounded domains.

The Boussinesq equations describe the motion of an incompressible viscous fluid subject to convective heat transfer. Decay rates of derivatives of solutions of the three-dimension-al Cauchy problem for a Boussinesq system are studied in this work.

Let $A_0$ be a closed, minimal symmetric operator from a Hilbert space $ℍ$ into $ℍ$ with domain not dense in $ℍ$. Let $\widehat{A}$ also be a correct selfadjoint extension of $A_0$. The purpose of this paper is (1) to characterize, with the help of $\widehat{A}$, all the correct selfadjoint extensions $B$ of $A_0$ with domain equal to $D\left(\widehat{A}\right)$, (2) to give the solution of their corresponding problems, (3) to find sufficient conditions for $B$ to be positive (definite) when $\widehat{A}$ is positive (definite).

We consider the internal stabilization of Maxwell's equations with Ohm's law with space variable coefficients in a bounded region with a smooth boundary. Our result is mainly based on an observability estimate, obtained in some particular cases by the multiplier method, a duality argument and a weakening of norm argument, and arguments used in internal stabilization of scalar wave equations.