Elliptic Operators with Unbounded Coefficients

Abstract

Aim of this manuscript is to give generation results and some Hardy inequalities concerning elliptic operators with unbounded coefficients of the form Au = div(aDu) + F ·Du + V u where V is a real valued function, a(x) = (akl(x)) is symmetric and satisfies the ellipticity condition and a and F grow to infinity. In particular, we mainly deal with Scrh¨odinger type operators, i.e., operators with vanishing drift term, ∇a + F = 0. The case of the whole operator is also considered in the sense that a weighted Hardy inequality for these operators is provided. Finally we will consider the higher order elliptic operator perturbed by a singular potential A = ∆2 −c|x|−4. Due to their importance for the strong relation with Schr¨odinger operators, we provide a survey on the most significant proofs of Hardy’s inequalities appeared in literature. Furthermore, we generalise Hardy inequality proving a weighted inequality with respect to a measure dµ = µ(x)dx satisfying suitable local integrability assumptions in the weighted spaces L2 µ(RN) = L2(RN,dµ). We claim that for all u ∈ H1 µ(RN), c ≤ c0,µ cZRN u2 |x|2 dµ ≤ZRN |∇u|2 dµ + CµZRN u2 dµ holds with c0,µ optimal constant. The interest in studying such an inequality is the relation with the parabolic problem associated to the Kolmogorov operator perturbed by a singular potential Lu = ∆u + ∇µ µ ·∇u + c |x|2 u. Moreover, we consider the Schr¨odinger type operator L0 with unbounded diffusion L0u = Lu + V u = (1 +|x|α)∆u + c |x|2 u with α ≥ 0 and c ∈R. The aim is to obtain sufficient conditions on the parameters ensuring that L0 with a suitable domain generates a quasi-contractive and positivity preserving C0-semigroup in Lp(RN), 1 < p < ∞. The proofs are based on some Lp-weighted Hardy inequality and perturbation techniques. In fact we treat the operator L0 as a perturbation of the elliptic operator L = (1+|x|α)∆ which has already been studied in literature. Finally, we study the biharmonic operator perturbed by an inverse fourthorder potential A = A0 −V = ∆2 − c |x|4 , 1 where c is any constant such that c < C∗ :=N(N−4) 4 2. Making use of the Rellich inequality, multiplication operators and off-diagonal estimates, we prove that the semigroup generated by −A in L2(RN), N ≥ 5, extrapolates to a bounded holomorphic C0-semigroup on Lp(RN) for all p ∈ [p0 0,p0], where p0 = 2N N−4 and p0 0 is its dual exponent. Furthermore, we study the boundedness of the Riesz transform ∆A−1/2 := 1 Γ(1/2)Z∞ 0 t−1/2∆e−tA dt on Lp(RN) for all p ∈ (p0 0,2]. The boundedness of ∆A−1/2 on Lp(RN) implies that the domain of A1/2 is included in the Sobolev space W2,p(RN). Thus, we obtain W2,p-regularity of the solution to the evolution equation with initial datum in Lp(RN) for p ∈ (p0 0,2], i.e., u(t) ∈ W2,p(RN). Publications [1] A. Canale, F. Gregorio, A. Rhandi, C. Tacelli: Weighted Hardy inequalities and Kolmogorov-type operators, preprint. [2] S. Fornaro, F. Gregorio, A. Rhandi: Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in Lp-spaces, Comm. on Pure and Appl. Anal. 15 (2016), no. 6, 2357-2372. [3] F. Gregorio, S. Mildner: Fourth-order Schr¨odinger type operator with singular potentials, Archiv der Mathematik 107 (2016), no. 3, 285-294. [edited by author]