By Guixia Pan, Lin Tang
We reflect on the Lp solvability for divergence and non-divergence shape Schrödinger equations with discontinuous coefficients. As an software, we provide the worldwide Morrey regularity for divergence and non-divergence shape Schrödinger operators with VMO coefficients in a bounded area.
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Additional resources for Solvability for Schrödinger equations with discontinuous coefficients
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By means of the deﬁnition of the coeﬃcients and the knowledge, now we have Mu(−x1 , x ) + λu(−x1 , x ) = −divg − f in Rn . accordingly, −u(−x1 , x ) is usually an answer to (5. 7). through the individuality of the answer, we receive u(x) = −u(−x1 , x ). this suggests that, as a functionality on Rn+ , u has 0 hint at the boundary and obviously u satisﬁes (5. 4). The life of the answer is proved. nonetheless, you can still see that if u ∈ W 1,p (Rn+ ) is an answer to (5. 4), then its bizarre extension with appreciate to x1 is an answer to (5.
19] A. Hinz, H. Kalf, Subsolution estimates and Harnack’s inequality for Schrödinger operators, J. Reine Angew. Math. 404 (1990) 118–131.  F. John, L. Nirenberg, On services of bounded suggest oscillation, Comm. natural Appl. Math. four (1961) 415–426.  D. Kim, N. Krylov, Elliptic diﬀerential equations with coeﬃcients measurable with appreciate to 1 variable and VMO with appreciate to the others, SIAM J. Math. Anal. 39 (2007) 489–506.  N. Krylov, Parabolic and elliptic equations with VMO coeﬃcients, Comm. Partial Diﬀerential Equations 32 (2007) 453–475.
The functionality m(x, V ) is deﬁned through ⎧ ⎪ ⎨ ρ(x) = 1 1 = sup r : n−2 m(x, V ) r>0 ⎪ r ⎩ ⎫ ⎪ ⎬ V (y)dy ≤ 1 B(x,r) ⎪ ⎭ . (2. 1) evidently, zero < m(x, V ) < ∞ if V = zero. specifically, m(x, V ) = 1 with V = 1 and m(x, V ) ∼ (1 + |x|) with V = |x|2 . Lemma 2. 1. (See . ) If V ∈ Bq for a few q ≥ such that n 2, then there exist l0 > zero and C0 > 1 m(x, V ) 1 −l l /(l +1) ≤ C0 (1 + |x − y|m(x, V )) zero zero (1 + |x − y|m(x, V )) zero ≤ . C0 m(y, V ) particularly, m(x, V ) ∼ m(y, V ) if |x − y| < C/m(x, V ). n Lemma 2.
Four. think that −a0ij Dij u + V u = zero in B(x0 , 2R) and V ∈ Bn/2 N ≥ zero, |∇2 u(x)| ≤ sup B(x0 ,2R) CN 1 . N [1 + Rm(x0 , V )] Rn 2 facts. allow η ∈ C0∞ (B(x0 , 2R)) such that η = 1 on B(x0 , 3R 2 ), |∇η| ≤ C/R and |∇ η| ≤ 2 C/R . be aware that, for x ∈ B(x0 , R), we have now Γ0 (x, y)(−a0ij Dij )(uη)(y)dy u(x)η(x) = Rn Γ0 (x, y)(−V uη − a0ij ∂i u∂j η − a0ij ∂j u∂i η − a0ij u∂ij η)dy = Rn Γ0 (x, y)(−V uη)dy − a0ij = Rn Γ0 (x, y)∂i u∂j ηdy Rn − a0ij Γ0 (x, y)∂j u∂i ηdy − a0ij Rn Γ0 (x, y)u∂ij ηdy, Rn the place Γ0 (x, y) denotes the basic resolution for the operator −a0ij Dij in Rn .