On self-adjointness of symmetric diffusion operators
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Journal of Evolution Equations
On self-adjointness of symmetric diffusion operators Derek W. Robinson
Abstract. Let be a domain in Rd with boundary and let d denote the Euclidean distance to . Further let H = − div(C∇) where C = ( ckl ) > 0 with ckl = clk real, bounded, Lipschitz continuous functions and D(H ) = Cc∞ (). The matrix Cd−δ is assumed to converge uniformly to a diagonal matrix a I as d → 0. Thus δ ≥ 0 measures the order of degeneracy of the operator and a, a positive Lipschitz function, gives the boundary profile of the operator. In addition we place a mild restriction on the order of degeneracy of the derivatives of the coefficients at the boundary. Then we derive sufficient conditions for H to be essentially self-adjoint as an operator on L 2 () in three general cases. Specifically, if is a C 2 -domain, or if = Rd \S where S is a countable set of positively separated points, or if = Rd \ with a convex set whose boundary has Hausdorff dimension d H ∈ {1, . . . , d − 1} then the condition δ > 2 − (d − d H )/2 is sufficient for essential self-adjointness. In particular δ > 3/2 suffices for C 2 -domains. Finally we prove that δ ≥ 3/2 is necessary in the C 2 -case.
1. Introduction Our intention is to analyze the L 2 -uniqueness of symmetric diffusion processes on a domain of the Euclidean space Rd with boundary . The problem can either be expressed as uniqueness of weak solutions of a diffusion equation ∂ϕt /∂t + H ϕt = 0 on L 2 () or, equivalently, as essential self-adjointness of the diffusion operator H = − div(C ∇) on Cc∞ (). In the standard theory of strongly elliptic operators uniqueness is ensured by the specification of conditions at the boundary. If, however, the coefficient matrix C is degenerate at the boundary then uniqueness is determined by the properties of the diffusion in a neighbourhood of . In both cases, however, the existence of a solution to the diffusion equation follows by specifying Dirichlet boundary conditions or, in operator terms, by constructing the self-adjoint Friedrichs’ extension HF of H . So in the degenerate case the uniqueness problem consists of relating the boundary behaviour to the uniqueness of the Dirichlet solution or the Friedrichs’ extension. Some guidance to L 2 -uniqueness is given by the better understood problem of L 1 uniqueness. The Friedrichs’ extension HF generates a submarkovian semigroup S F on L 2 (), i.e. a semigroup which is both positive and L ∞ -contractive, and this leads to a Mathematics Subject Classification: 31C25, 47D07 Keywords: Self-adjointness, L 1 -uniqueness, Diffusion operators, Rellich inequalities.
D. W. Robinson
J. Evol. Equ.
natural probabilistic interpretation. Then L 1 -uniqueness is equivalent to S F conserving probability by Theorem 2.2 in [8] and this in turn is equivalent to HF being the unique self-adjoint extension of H which generates a submarkovian semigroup (see [10], Corollary 3.4, and [29], Theorem 1.3). Thus the L 1 -problem reduces to an L 2 -problem, Markov uniqueness, which turns out
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