# The Cauchy problem (constant coefficients)

• Lars Hörmander
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 116)

## Abstract

To solve the Cauchy problem for a differential operator P (D) with data on a plane 〈x, N〉 = 0, where 0 ⧧ N Є R n , means, roughly speaking, to find a solution u of the equation
$$% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacI % cacaWGebGaaiykaiaadwhacaqGGaGaeyypa0JaaeiiaiaadAgacaGG % Saaaaa!3DCC! P(D)u{\text{ }} = {\text{ }}f,$$
(5.0.1)
where f is given, so that for another given function φ
$$% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWG1bGaeyOeI0Iaeqy1dyMaeyypa0JaaGimamaabmaapaqaa8qa % daaadaWdaeaapeGaamiEaiaacYcacaWGobaacaGLPmIaayPkJaWdam % aaCaaaleqabaWdbiaad2gaaaaakiaawIcacaGLPaaacaqGGaGaam4D % aiaadIgacaWGLbGaamOBaiaabccadaaadaWdaeaapeGaamiEaiaacY % cacaWGobaacaGLPmIaayPkJaGaeyOKH4QaaGimaiaac6caaaa!4FB6! u - \phi = 0\left( {{{\left\langle {x,N} \right\rangle }^m}} \right){\text{ }}when{\text{ }}\left\langle {x,N} \right\rangle \to 0.$$
(5.0.2)

## Keywords

Cauchy Problem Differential Operator Fundamental Solution Half Space Constant Coefficient
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.