Abstract. We study, with purely analytic tools, existence, uniqueness and gradient estimates of the solutions to the Neumann problems associated with a second order elliptic operator with unbounded coefficients in spaces of continuous functions in an unbounded open set Ω in R N .
Let a and b be unbounded functions in R N with a sufficiently smooth. In this paper, we prove that, under suitable growth assumptions on a and b, the operator Au = a∆u + b • ∇u admits realizations generating analytic semigroups in L p (R N ) for any p ∈ [1, +∞] and in C b (R N ). We also explicitly characterize the domain of the infinitesimal generator of such semigroups. Similar results are stated and proved when R N is replaced by a smooth exterior domain under general boundary conditions.
We consider autonomous parabolic Dirichlet problems in a regular unbounded open set ⊂ R N involving second-order operator A with (possibly) unbounded coefficients. We determine new conditions on the coefficients of A yielding global gradient estimates for the bounded classical solution.
We consider a class of uniformly elliptic operators A with unbounded coe‰cients in unbounded domains W. Under suitable assumptions on the geometry of W and on the co-e‰cients, we prove that the Cauchy-Neumann problem associated with the operator A admits a unique bounded classical solution u for any initial datum f which is bounded and continuous in W. Moreover, we prove uniform and pointwise gradient estimates for u. Finally, we give some applications of the so obtained estimates.
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