Gunnar Aronsson

Extension of functions satisfying lipschitz conditions

Le t F be a compac t set in Euc l idean n-space RL Suppose t h a t the funct ion ~(P) is def ined on F and t h a t i t satisfies a Lipschi tz condit ion. 1 I t is known t h a t ~0 can be e x t e n d e d to R ~ in such a w a y t h a t the new funct ion / satisfies t he same Lipsehi tz condit ion. An expl ic i t cons t ruc t ion was given in [1], where even more genera l s i tuat ions were t r ea t ed . I t is ea sy to give uppe r and lower bounds for the solutions. W e shal l do this, and also discuss quest ions of uniqueness. I t t u rns out t h a t a l l solut ions agree on a set E which has a ve ry s imple s t ruc ture (Theorem 2). I n ana logy wi th the t r e a t m e n t in [2] a n d [3], we shal l consider a subclass of solut ions (called absolu te minimals) which have the add i t i ona l p r o p e r t y of being solut ions of a corresponding p rob lem on each subregion of R ~ F . The pa r t i a l d i f ferent ia l equa t ion ~ . j=l, tx~ r tx~xj = 0 is de r ived in a pu re ly fo rmal m a n n e r and i t t u rns out t h a t a smooth funct ion satisfies th is equa t ion if and on ly if i t is an absolu te min ima l (Theorem 8). W e shal l also give an exis tence proof for abso lu te minima]s. I n a l a t e r paper , the two-d imens iona l case, a n d in pa r t i cu l a r the d i f ferent ia l equat ion r = 0, more r162 r tzu +Ca tuu will be s tud ied closely.

A deterministic model in biomathematics. asymptotic behavior and threshold conditions

Abstract We study a deterministic SIS model, which takes into account seasonal variations as well as heterogeneities in the population. The appropiaate threshold condition is found (the condition for the disease to remain endemic).

Perfect splines and nonlinear optimal control theory

Abstract In this paper we use a method from nonlinear optimal control theory to establish the “perfect spline” properties of a solution to a certain extremum problem. The problem is to minimize the L ∞ norm of a nonlinear expression of the form F(t, x(t), x (t), x (t),…, x (n) (t)) over all sufficiently smooth functions x ( t ) which satisfy given boundary conditions. Under suitable assumptions, we show that a solution x 0 ( t ) must be such that F(t, x 0 (t), x 0 (t),…, x 0 (n) (t)) is constant, and x 0 ( n ) ( t ) is piece-wise continuous with a finite number of jump discontinuities. This generalizes results by D. S. Carter, G. Glaeser, D. McClure, and others, who studied the same problem for linear differential expressions.

Linear control theory applied to a minimum-maximum problem

A linearisation technique and well-established linear control theory are used to derive relevant information concerning the extremal functions in a minimum-maximum problem