We investigate the shape of the solution of the Cauchy problem for the damped wave equation. In particular, we study the existence, location and number of spatial maximizers of the solution.Studying the shape of the solution of the damped wave equation, we prepare a decomposed form of the solution into the heat part and the wave part. Moreover, as its another application, we give L p -L q estimates of the solution.

We investigate a potential with a radially symmetric and strictly decreasing kernel depending on a parameter. We regard the potential as a function defined on the upper half-space R m × (0, +∞) and study some geometric properties of its spatial maximizer. To be precise, we give some sufficient conditions for the uniqueness of a maximizer of the potential and study the asymptotic behavior of the set of maximizers.Using these results, we imply geometric properties of some specific potentials. In particular, we consider applications for the solution of the Cauchy problem for the heat equation, the Poisson integral (including a solid angle) and r α−m -potentials.

The object of our investigation is a point that gives the maximum value of a potential with a strictly decreasing radially symmetric kernel. It defines a center of a body in R m . When we choose the Riesz kernel or the Poisson kernel as the kernel, such centers are called an r α−m -center or an illuminating center, respectively.The existence of a center is easily shown but the uniqueness does not always hold. Sufficient conditions of the uniqueness of a center have been studied by some researchers. The main results in this paper are some new sufficient conditions for the uniqueness of a center of a body. , 26B25. points of the line segments OP , P Q and QO, respectively. We remark that the minimal unfolded region of Ω is contained in △ABC (see Example 2.5).We identify the notation z j for the j-th coordinate with the function z j : R 2 ∋ (z 1 , z 2 ) → z j ∈ R. We denote the point R θ P by P θ , for short, and so on.We have to consider the following eleven cases about the position of R θ Ω (see Figure 9 to 19):Case I The rotation angle θ is non-negative.and the slope of the line P θ Q θ is non-positive. Case I.3.2 0 ≤ z 1 (B θ ) ≤ z 1 (A θ ) and the slope of the line P θ Q θ is non-negative. Case I.4.1 z 1 (B θ ) ≤ 0 ≤ z 1 (A θ ) and the slope of the line P θ Q θ is non-positive. Case I.4.2 z 1 (B θ ) ≤ 0 ≤ z 1 (A θ ) and the slope of the line P θ Q θ is non-negative.Case II The rotation angle θ is non-positive.Case II.2.1 z 1 (A θ ) ≤ z 1 (C θ ) ≤ z 1 (P θ ) and the slope of the line OQ θ is non-negative.Case II.2.2 z 1 (A θ ) ≤ z 1 (C θ ) ≤ z 1 (P θ ) and the slope of the line OQ θ is non-positive.Case II.3.1 z 1 (P θ ) ≤ z 1 (C θ ) and the slope of the line OQ θ is non-negative.Case II.3.2 z 1 (P θ ) ≤ z 1 (C θ ) and the slope of the line OQ θ is non-positive.

A space curve is determined by conformal arc-length, conformal curvature, and conformal torsion, up to Möbius transformations. We use the spaces of osculating circles and spheres to give a conformally defined moving frame of a curve in the Minkowski space, which can naturally produce the conformal invariants and the normal form of the curve. We also give characterization of canal surfaces in terms of curves in the set of circles.

We consider a projection from the center of the unit sphere to a tangent space of it, the central projection, and study two area minimizing problems of the image of a closed subset in the sphere. One of the problems is the uniqueness of the tangent plane that minimizes the area for an arbitrary fixed subset. The other is the shape of the subset that minimizes the minimum value of the area. We also study the similar problems for the hyperbolic space.