Uniqueness of diffusion operators and capacity estimates

作者：Robinson Derek W^*

来源：Journal of Evolution Equations, 2013, 13(1): 229-250.

DOI：10.1007/s00028-013-0176-4

Let Omega be a connected open subset of R d . We analyse L 1-uniqueness of real second-order partial differential operators H = -Sigmak,l=1d partial derivativekcklpartial derivativel and K = H + Sigmak=1d ck partial derivativek + c0 on Omega where ckl = clk is an element of Wloc1,infinity(Omega), ck is an element of Linfinity,loc(omega), c0 is an element of L2,loc(Omega) and C(x) = (ckl(x))> 0 for all x is an element of omega. Boundedness properties of the coefficients are expressed indirectly in terms of the balls B(r) associated with the Riemannian metric C -1 and their Lebesgue measure vertical bar B(r)vertical bar. First, we establish that if the balls B(r) are bounded, the Tacklind condition integralRinfinitydr r(log vertical bar B(r)vertical bar)-1 = infinity is satisfied for all large R and H is Markov unique then H is L 1-unique. If, in addition, C(x)>= k(cT circle times c)(x)for some k > 0 and almost all x is an element of omega div c is an element of Linfinity,1oc,(omega) is upper semi-bounded and c 0 is lower semi-bounded, then K is also L 1-unique. Secondly, if the c kl extend continuously to functions which are locally bounded on partial derivative Omega and if the balls B(r) are bounded, we characterize Markov uniqueness of H in terms of local capacity estimates and boundary capacity estimates. For example, H is Markov unique if and only if for each bounded subset A of (Omega)over-bar there exist etan is an element of Ccinfinity(Omega)satisfying limn similar to infinityparallel to 1AGamma(etan)parallel to1 = 0, where Gamma(etan) = Sigmak,ld=1ckl(partial derivativeketan)(partial derivativeletan) and limnsimilar to infinity parallel to 1A(1Omega - etan)phi parallel to2 = 0 for each phi is an element of L2(Omega) or if and only if cap(partial derivative Omega) = 0.

• 出版日期2013-3