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Elliptic and parabolic secondorder PDEs with growing coefficients
"... Abstract. We consider a secondorder parabolic equation in R d+1 with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally Hölder continuous in the space variables. We show that global Schauder estimates hold even in this ca ..."
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Abstract. We consider a secondorder parabolic equation in R d+1 with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally Hölder continuous in the space variables. We show that global Schauder estimates hold even in this case. The proof introduces a new localization procedure. Our results show that the constant appearing in the classical Schauder estimates is in fact independent of the L∞norms of the lower order coefficients. We also give a proof of uniqueness which is of independent interest even in the case of bounded coefficients.
Maximal regularity in L p (R N ) for a class of elliptic operators with unbounded coefficients, Differential Integral Equations
 Di¤ erential Integral Equations
"... Abstract. Strongly elliptic differential operators with (possibly) unbounded lower order coefficients are shown to generate C0semigroups on L p (R N), 1 < p < +∞. An explicit characterization of the domain is given. 1. ..."
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Abstract. Strongly elliptic differential operators with (possibly) unbounded lower order coefficients are shown to generate C0semigroups on L p (R N), 1 < p < +∞. An explicit characterization of the domain is given. 1.
Discreteness of the Spectrum for Some Differential Operators With Unbounded Coefficients in R^n
"... We give sufficient conditions for the discreteness of the spectrum of differential operators of the form Au = \Gamma\Deltau +hrF;rui in L 2 (R n ) where d(x) = e \GammaF (x) dx and for Schrödinger operators in L 2 (R n ). Our conditions are also necessary in the case of polynomial coeffic ..."
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We give sufficient conditions for the discreteness of the spectrum of differential operators of the form Au = \Gamma\Deltau +hrF;rui in L 2 (R n ) where d(x) = e \GammaF (x) dx and for Schrödinger operators in L 2 (R n ). Our conditions are also necessary in the case of polynomial coefficients.
SCUOL A NORM ALE SUPER IORE Pisa
"... The Neu man n prob lem on unb oun ded dom ains of R^d and stoc has tic vari atio nal ineq uali ties ..."
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The Neu man n prob lem on unb oun ded dom ains of R^d and stoc has tic vari atio nal ineq uali ties
unknown title
, 2005
"... This article was originally published in a journal published by Elsevier, and the attached copy is provided by Elsevier for the author’s benefit and for the benefit of the author’s institution, for noncommercial research and educational use including without limitation use in instruction at your in ..."
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This article was originally published in a journal published by Elsevier, and the attached copy is provided by Elsevier for the author’s benefit and for the benefit of the author’s institution, for noncommercial research and educational use including without limitation use in instruction at your institution, sending it to specific colleagues that you know, and providing a copy to your institution’s administrator. All other uses, reproduction and distribution, including without limitation commercial reprints, selling or licensing copies or access, or posting on open internet sites, your personal or institution’s website or repository, are prohibited. For exceptions, permission may be sought for such use through Elsevier’s permissions site at:
EXISTENCE AND UNIQUENESS TO THE CAUCHY PROBLEM FOR LINEAR AND SEMILINEAR PARABOLIC EQUATIONS WITH LOCAL CONDITIONS. ∗
"... Abstract. We consider the Cauchy problem in Rd for a class of semilinear parabolic partial differential equations that arises in some stochastic control problems. We assume that the coefficients are unbounded and locally Lipschitz, not necessarily differentiable, with continuous data and local unif ..."
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Abstract. We consider the Cauchy problem in Rd for a class of semilinear parabolic partial differential equations that arises in some stochastic control problems. We assume that the coefficients are unbounded and locally Lipschitz, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution by approximation with linear parabolic equations. The linear equations involved can not be solved with the traditional results. Therefore, we construct a classical solution to the linear Cauchy problem under the same hypotheses on the coefficients for the semilinear equation. Our approach is using stochastic differential equations and parabolic differential equations in bounded domains. Finally, we apply the results to a stochastic optimal consumption problem. Résumé. Nous considérons le problème de Cauchy dans Rd pour une classe d’équations aux dérivées partielles paraboliques semi linéaires qui se pose dans certains problèmes de contrôle stochastique. Nous supposons que les coefficients ne sont pas bornés et sont localement Lipschitziennes, pas nécessairement différentiables, avec des données continues et ellipticite ́ local uniforme. Nous construisons une solution classique par approximation avec les équations paraboliques linéaires. Les équations linéaires impliquées ne peuvent être résolues avec les résultats traditionnels. Par conséquent, nous construisons une solution classique au problème de Cauchy linéaire sous les mêmes hypothèses sur les coefficients pour l’équation semilinéaire. Notre approche utilise les équations différentielles stochastiques et les équations différentielles paraboliques dans les domaines bornés. Enfin, nous appliquons les résultats a ̀ un problème stochastique de consommation optimale.
Regularity properties for second order partial differential operators with unbounded coefficients
"... It is a great pleasure to express my warmest thanks to some people whose support and help have been fundamental for the realization of this thesis. First of all I sincerely thank my supervisor, Prof. Giorgio Metafune, who followed my work with interest and patience, teaching me a method of research ..."
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It is a great pleasure to express my warmest thanks to some people whose support and help have been fundamental for the realization of this thesis. First of all I sincerely thank my supervisor, Prof. Giorgio Metafune, who followed my work with interest and patience, teaching me a method of research and infecting me with his great enthusiasm for mathematics. I am greatful to Diego Pallara, who often helped and encouraged me during these years and to Vincenzo Vespri, for useful conversations. I owe my gratitude to my colleagues and friends Marcello Bertoldi, Giovanni Cupini, Vincenzo Manco, Enrico Priola who have accompanied the progress of my research very patiently, sharing their time and their ideas with me. I wish to dedicate this work to my grandfather, who has taught me a great love for life and whose memory will remain always alive in my heart.