A priori inequalities connected with systems of partial ... - Project Euclid
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A priori inequalities connected with systems of partial ... - Project Euclid
find a necessary and sufficient condition for the existence of a constant C ... decide under which conditions an inequality of the above type holds for all .... we choose an infinitely differentiable, non-zero function k = ]c (t) of a single real var- .... Before describing the second case of a priori inequalities to be considered in the.
A PRIORI INEQUALITIES CONNECTED WITH SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS BY
BENT FUGLEDE Copenhagen
Introduction In the more recent development of the theory of general (linear) partial differential equations, the so-called a priori inequalities play a prominent role. For instanc% the important comparison of two partial differential operators P (D) and Q (D) witht constant coefficients, studied b y L. HSrmandcr [1], depends on the existence of a constant C such t h a t
II Q (D) u l] 0, provided P (~) * 0. It is solely the order of magnitude of the norm function which matters in the applications to differential operators. Accordingly, /5(~) could be replaced by any function F (~) equivalent to P (~) in the sense that
c-~ ,5 (8) < F (8) < C_~ (~)
(3)
for some constant C. For instance, ~ ] P(~)(~)l and max~ [P(~))~)] are equivalent to /5(~). The following lemma contains further examples of such functions. We denote b y C ~ the complex n-dimensional space and write ]~]2= ]~112+ ... + ] ~ p
when
= (~,. . . . . ~.) c c". L]~MMA 1.1. Each o/ the /ollowing /unctions is equivalent to /5(~):
