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

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 $C_{b}(\overline{\O})$.
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
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