A convolution of Stieltjes type is introduced for operator families of bounded strong variation and vector valued $L_1$-functions. Using this tool, perturbation theorems for integrated semigroups of bounded strong variation are derived, and improved results on $L_p$-solutions to the inhomogeneous Cauchy problem are obtained.

We study the existence, uniqueness and stability of global strong solutions to the mixed problem for the nonhomogeneous Kirchhoff equation $$ u_{tt}(x,t) - \varphi (x,t) M \Big (\int_{\Omega} \mid \nabla u(\xi,t) \mid^{2}\, d\xi \Big ) \Delta u(x,t) + g(x, t, u_{t}(x,t)) = 0 $$ with small initial data.

This paper is devoted to the systematic investigation of the completeness of root vectors (generalized eigenvectors) of a non-selfadjoint Agmon-Douglis-Nirenberg (ADN) elliptic boundary problem with an indefinite weight matrix. As it is known the realization of a ADN elliptic problem (or simply a correctly posed general elliptic boundary problem in which no restrictions are imposed on the order of some boundary operators) is not in general densely defined. This amounts to some tremendous difficulties in the derivation of the completeness of the root vectors for the corresponding spectral problem. However under some circumstances the completeness of root vectors may be obtained in some dense subsets of functions in Sobolev spaces. These issues are addressed in the present work for a class of non-selfadjoint ADN elliptic boundary value problems with an indefinite weight matrix-function for which we establish new completeness results for the corresponding root vectors in appropriate $L_{p}$ Sobolev spaces.

We study the regularity of propagating fronts whose motion is anisotropic. We prove that there is at most one normal direction at each point of the front; as an application, we prove that convex fronts are $C^{1,1}.$ These results are by-products of some necessary conditions for viscosity solutions of quasilinear elliptic equations. These conditions are of independent interest; for instance they imply some regularity for viscosity solutions of nondegenerate quasilinear elliptic equations.

We discuss the boundary value problem $\Delta u=f$, $\partial^ku/\partial n^k=g$ in a bounded two-dimensional domain. For a smooth simply connected region, we prove that the only solutions of the homogeneous problem are harmonic polynomials of degree $k-1$. For multiply connected domains, this is true ``generically." In domains with corners, on the other hand, there are additional solutions.