Remarks on variational principles in elastodynamics
Two principles are basic in the variational formulations of elasticity. Green's “variation for displacements” leads to the principle of minimum potential energy, whereas Castigliano's “variation for stresses” leads to the principle of minimum complementary energy. In the small vibration of elastic bodies, the result of Green's approach coincides with Hamilton's principle of dynamics. Toupin (3) and Crandall (4) both established complementary extremum principles for dynamics. Reissner (1) has derived a complementary extremum principle for steady-state vibration of elastic bodies. The present paper derives the extremum principle for small vibration of elastic bodies in general.

title: On the Vibration of Beams or Rods Carrying a Concentrated Mass

title: Der Satz von Miquel in der Möbiusebene

title: Partial regularity for weak heat flows into spheres
abstract: In this paper we show that a weak heat flow of harmonic maps from a compact Riemannian manifold (possibly with boundary) into a sphere, satisfying the monotonicity inequality and the energy inequality, is regular off a closed set of m-dimensional Hausdorff measure zero.

title: Evolution equations with a free boundary condition
abstract: In this paper we consider the heat flow of harmonic maps between two compact Riemannian Manifolds M and N (without boundary) with a free boundary condition. That is, the following initial boundary value problem ∂1,u −Δu = Γ(u)(∇u, ∇u) [tT TuuN, on M × [0, ∞), u(t, x) ∈ Σ, for x ∈ ∂M, t > 0, ∂u/t6n(t, x) ⊥u Tu(t,x) Σ, for x ∈ ∂M, t > 0, u(o,x) = uo(x), on M, where Σ is a smooth submanifold without boundary in N and n is a unit normal vector field of M along ∂M.Due to the higher nonlinearity of the boundary condition, the estimate near the boundary poses considerable difficulties, even for the case N = ℝn, in which the nonlinear equation reduces to ∂tu-Δu = 0.We proved the local existence and the uniqueness of the regular solution by a localized reflection method and the Leray-Schauder fixed point theorem. We then established the energy monotonicity formula and small energy regularity theorem for the regular solutions. These facts are used in this paper to construct various examples to show that the regular solutions may develop singularities in a finite time. A general blow-up theorem is also proven. Moreover, various a priori estimates are discussed to obtain a lower bound of the blow-up time. We also proved a global existence theorem of regular solutions under some geometrical conditions on N and Σ which are weaker than KN <-0 and Σ is totally geodesic in N.

title: Initial value problems for nonlinear heat equations

title: On existence theorems of periodic traveling wave solution to the generalized forced Kadomtsev-Petviashvili equation
abstract: T uts PAPER is concerned with periodic travcling wave solutions of the generalized forced KadomtsevPetviashvili equation in the form ( u 1 + If( tt)lx + o u x.rx )"' + {3uyy = ho. The basic approach to this problem is to establish an equivalence relat ionship between a periodic boundary value problem and nonlinear integral equations wirh symrnerric kernels by using the Green's function method. The integral representations generate compact operators in a Banach space of real-valued continuous periodic functions with a given period 2T. Schauder's fixed point theorem is then used to prove the existence of no nconstant periodic traveling wave solutions.