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Regularity theory and Green’s function for elliptic equations with lower order terms in unbounded domains

2023; Springer Science+Business Media; Volume: 62; Issue: 9 Linguagem: Inglês

10.1007/s00526-023-02596-2

1432-0835

Mihalis Mourgoglou,

Numerical methods in inverse problems

Abstract We consider elliptic operators in divergence form with lower order terms of the form $$Lu = -{{\textrm{div}}}(A \cdot \nabla u + b u ) - c \cdot \nabla u - du$$ L u = - div ( A · ∇ u + b u ) - c · ∇ u - d u , in an open set $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n , $$n \ge 3$$ n ≥ 3 , with possibly infinite Lebesgue measure. We assume that the $$n \times n$$ n × n matrix A is uniformly elliptic with real, merely bounded and possibly non-symmetric coefficients, and either $$b, c \in L^{n,\infty }_{\text {loc}}({\Omega })$$ b , c ∈ L loc n , ∞ ( Ω ) and $$d \in L_{\text {loc}}^{\frac{n}{2}, \infty }(\Omega )$$ d ∈ L loc n 2 , ∞ ( Ω ) , or $$|b|^2, |c|^2, |d| \in \mathcal {K}_{\text {loc}}(\Omega )$$ | b | 2 , | c | 2 , | d | ∈ K loc ( Ω ) , where $$\mathcal {K}_{\text {loc}}(\Omega )$$ K loc ( Ω ) stands for the local Stummel–Kato class. Let $${\mathcal {K}_{\text {Dini}}}(\Omega )$$ K Dini ( Ω ) be a variant of $$\mathcal {K}(\Omega )$$ K ( Ω ) satisfying a Carleson-Dini-type condition. We develop a De Giorgi/Nash/Moser theory for solutions of $$Lu = f - {{\textrm{div}}}g$$ L u = f - div g , where f and $$|g|^2 \in {\mathcal {K}_{\text {Dini}}}(\Omega )$$ | g | 2 ∈ K Dini ( Ω ) if, for $$q \in [n, \infty )$$ q ∈ [ n , ∞ ) , any of the following assumptions holds: (i) $$|b|^2, |d| \in {\mathcal {K}_{\text {Dini}}}(\Omega )$$ | b | 2 , | d | ∈ K Dini ( Ω ) and either $$c \in L^{n,q}_{\text {loc}}(\Omega )$$ c ∈ L loc n , q ( Ω ) or $$|c|^2 \in \mathcal {K}_{\text {loc}}(\Omega )$$ | c | 2 ∈ K loc ( Ω ) ; (ii) $${{\textrm{div}}}b +d \le 0$$ div b + d ≤ 0 and either $$b+c \in L^{n,q}_{\text {loc}}(\Omega )$$ b + c ∈ L loc n , q ( Ω ) or $$|b+c|^2 \in \mathcal {K}_{\text {loc}}(\Omega )$$ | b + c | 2 ∈ K loc ( Ω ) ; (iii) $$-{{\textrm{div}}}c + d \le 0$$ - div c + d ≤ 0 and $$|b+c|^2 \in {\mathcal {K}_{\text {Dini}}}(\Omega )$$ | b + c | 2 ∈ K Dini ( Ω ) . We also prove a Wiener-type criterion for boundary regularity. Assuming global conditions on the coefficients, we show that the variational Dirichlet problem is well-posed and, assuming $$-{{\textrm{div}}}c +d \le 0$$ - div c + d ≤ 0 , we construct the Green’s function associated with L satisfying quantitative estimates. Under the additional hypothesis $$|b+c|^2 \in \mathcal {K}'(\Omega )$$ | b + c | 2 ∈ K ′ ( Ω ) , we show that it satisfies global pointwise bounds and also construct the Green’s function associated with the formal adjoint operator of L . An important feature of our results is that all the estimates are scale invariant and independent of $$\Omega $$ Ω , while we do not assume smallness of the norms of the coefficients or coercivity of the associated bilinear form.