American Institute of Mathematical Sciences

Advanced
December  2008, 22(4): 933-953. doi: 10.3934/dcds.2008.22.933

On a class of elliptic and parabolic equations in convex domains without boundary conditions

Giuseppe Da Prato 1, and  Alessandra Lunardi 2,

1. 

Scuola Normale Superiore, Piazza dei Cavalieri 6, 56126 Pisa, Italy
2. 

Dipartimento di Matematica, Università di Parma, Viale G. Usberti 85/A, 43100 Parma

Received  August 2007 Published  September 2008

We consider the operator $\A u = \frac{1}{2} \Delta u - \langle DU, Du\right$, where $U $ is a convex real function defined in a convex open set $\O \subset \R^N$ and $\lim_{|x|\to \infty} U(x) = \lim_{ x \to \partial \O} U(x)$ $ =$ $ +\infty$. We study the realization of $\A $ in the spaces $C_{b}(\overline{\O})$, $C_{b}(\O)$ and $B_{b}(\O)$, and prove several properties of the associated Markov semigroup. In contrast with the case of bounded coefficients, elliptic equations and parabolic Cauchy problems such as (3) and (4) below are uniquely solvable in reasonable classes of functions, without imposing any boundary condition. We prove that the associated semigroup coincides with the transition semigroup of a stochastic variational inequality on $C_{b}(\overline{\O})$.
Keywords: Markov semigroups, stochastic variational inequalities, unbounded coefficients.
Mathematics Subject Classification: Primary: 47D07, 60J35; Secondary: 35J99, 35K99.
Citation: Giuseppe Da Prato, Alessandra Lunardi. On a class of elliptic and parabolic equations in convex domains without boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 933-953. doi: 10.3934/dcds.2008.22.933
[1]

Xiaojun Chen, Guihua Lin. CVaR-based formulation and approximation method for stochastic variational inequalities. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 35-48. doi: 10.3934/naco.2011.1.35
[2]

Rui Mu, Zhen Wu. Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs. Mathematical Control & Related Fields, 2017, 7 (2) : 289-304. doi: 10.3934/mcrf.2017010
[3]

Luigi Greco, Gioconda Moscariello, Teresa Radice. Nondivergence elliptic equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 131-143. doi: 10.3934/dcdsb.2009.11.131
[4]

Paulo Cesar Carrião, Olimpio Hiroshi Miyagaki. On a class of variational systems in unbounded domains. Conference Publications, 2001, 2001 (Special) : 74-79. doi: 10.3934/proc.2001.2001.74
[5]

Hongxia Yin. An iterative method for general variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 201-209. doi: 10.3934/jimo.2005.1.201
[6]

R. Bartolo, Anna Maria Candela, J.L. Flores. Timelike Geodesics in stationary Lorentzian manifolds with unbounded coefficients. Conference Publications, 2005, 2005 (Special) : 70-76. doi: 10.3934/proc.2005.2005.70
[7]

Giorgio Metafune, Chiara Spina, Cristian Tacelli. On a class of elliptic operators with unbounded diffusion coefficients. Evolution Equations & Control Theory, 2014, 3 (4) : 671-680. doi: 10.3934/eect.2014.3.671
[8]

Bálint Farkas, Luca Lorenzi. On a class of hypoelliptic operators with unbounded coefficients in $R^N$. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1159-1201. doi: 10.3934/cpaa.2009.8.1159
[9]

Katarzyna Pichór, Ryszard Rudnicki. Applications of stochastic semigroups to cell cycle models. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2365-2381. doi: 10.3934/dcdsb.2019099
[10]

Jerome A. Goldstein, Rosa Maria Mininni, Silvia Romanelli. Generators of Feller semigroups with coefficients depending on parameters and optimal estimators. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 511-527. doi: 10.3934/dcdsb.2007.8.511
[11]

Shige Peng, Mingyu Xu. Constrained BSDEs, viscosity solutions of variational inequalities and their applications. Mathematical Control & Related Fields, 2013, 3 (2) : 233-244. doi: 10.3934/mcrf.2013.3.233
[12]

Michel Chipot, Karen Yeressian. On the asymptotic behavior of variational inequalities set in cylinders. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4875-4890. doi: 10.3934/dcds.2013.33.4875
[13]

Qingzhi Yang. The revisit of a projection algorithm with variable steps for variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 211-217. doi: 10.3934/jimo.2005.1.211
[14]

P. Smoczynski, Mohamed Aly Tawhid. Two numerical schemes for general variational inequalities. Journal of Industrial & Management Optimization, 2008, 4 (2) : 393-406. doi: 10.3934/jimo.2008.4.393
[15]

Lori Badea. Multigrid methods for some quasi-variational inequalities. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1457-1471. doi: 10.3934/dcdss.2013.6.1457
[16]

G. Mastroeni, L. Pellegrini. On the image space analysis for vector variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (1) : 123-132. doi: 10.3934/jimo.2005.1.123
[17]

Yusuke Murase, Risei Kano, Nobuyuki Kenmochi. Elliptic Quasi-variational inequalities and applications. Conference Publications, 2009, 2009 (Special) : 583-591. doi: 10.3934/proc.2009.2009.583
[18]

G. Idone, A. Maugeri. Variational inequalities and a transport planning for an elastic and continuum model. Journal of Industrial & Management Optimization, 2005, 1 (1) : 81-86. doi: 10.3934/jimo.2005.1.81
[19]

Barbara Panicucci, Massimo Pappalardo, Mauro Passacantando. On finite-dimensional generalized variational inequalities. Journal of Industrial & Management Optimization, 2006, 2 (1) : 43-53. doi: 10.3934/jimo.2006.2.43
[20]

Dimitri Mugnai. Almost uniqueness result for reversed variational inequalities. Conference Publications, 2007, 2007 (Special) : 751-757. doi: 10.3934/proc.2007.2007.751

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences