Let L be a second order elliptic operator on with a constant diffusion matrix and a dissipative (in a weak sense) drift with some . We assume that L possesses a Lyapunov function, but no local boundedness of b is assumed. It is known that then there exists a unique probability measure μ satisfying the equation and that the closure of L in generates a Markov semigroup with the resolvent . We prove that, for any Lipschitzian function and all , the functions and are Lipschitzian and and . In addition, we show that for every bounded Lipschitzian function g, the function is the unique bounded solution of the equation in the Sobolev class .
Soit L un opérateur elliptique sur tel que son terme du premier ordre , , soit dissipatif (mais pas nécessairement localement borné) et qu'il existe une fonction de Liapounoff. Il est connu qu'il existe une probabilité unique μ telle que au sens faible et la fermeture de L dans est le générateur d'un semigroupe markovien de résolvante . Nous montrons que pour chaque fonction lipschitzienne et tous les fonctions et sont lipschitziennes et on a et . De plus, nous montrons que pour chaque fonction bornée lipschitzienne g la fonction est la solution unique bornée de l'équation dans la classe de Sobolev .
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Vladimir I. Bogachev 1; Giuseppe Da Prato 2; Michael Röckner 3; Zeev Sobol 4
@article{CRMATH_2004__339_4_277_0, author = {Vladimir I. Bogachev and Giuseppe Da Prato and Michael R\"ockner and Zeev Sobol}, title = {Global gradient bounds for dissipative diffusion operators}, journal = {Comptes Rendus. Math\'ematique}, pages = {277--282}, publisher = {Elsevier}, volume = {339}, number = {4}, year = {2004}, doi = {10.1016/j.crma.2004.05.016}, language = {en}, }
TY - JOUR AU - Vladimir I. Bogachev AU - Giuseppe Da Prato AU - Michael Röckner AU - Zeev Sobol TI - Global gradient bounds for dissipative diffusion operators JO - Comptes Rendus. Mathématique PY - 2004 SP - 277 EP - 282 VL - 339 IS - 4 PB - Elsevier DO - 10.1016/j.crma.2004.05.016 LA - en ID - CRMATH_2004__339_4_277_0 ER -
Vladimir I. Bogachev; Giuseppe Da Prato; Michael Röckner; Zeev Sobol. Global gradient bounds for dissipative diffusion operators. Comptes Rendus. Mathématique, Volume 339 (2004) no. 4, pp. 277-282. doi : 10.1016/j.crma.2004.05.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.05.016/
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