scholarly journals Higher-Order Elliptic and Parabolic Equations with VMO Assumptions and General Boundary Conditions

2019 ◽  
Author(s):  
Hongjie Dong ◽  
Chiara Gallarati
Keyword(s):  
Boundary Conditions ◽  
Parabolic Equations ◽  
Higher Order ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Elliptic And Parabolic Equations

scholarly journals Higher-Order Parabolic Equations with VMO Assumptions and General Boundary Conditions with Variable Leading Coefficients

2018 ◽  
Vol 2020 (7) ◽  
pp. 2114-2144 ◽  
Author(s):  
Hongjie Dong ◽  
Chiara Gallarati
Keyword(s):  
Boundary Conditions ◽  
Half Space ◽  
Parabolic Equations ◽  
Higher Order ◽  
Elliptic Operators ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Elliptic And Parabolic Equations ◽  
Boundary Operators ◽  
Higher Order Parabolic Equations

Abstract We prove weighted mixed $L_{p}(L_{q})$-estimates, with $p,q\in (1,\infty )$, and the corresponding solvability results for higher-order elliptic and parabolic equations on the half space ${\mathbb{R}}^{d+1}_{+}$ and on general $C^{2m-1,1}$ domains with general boundary conditions, which satisfy the Lopatinskii–Shapiro condition. We assume that the elliptic operators A have leading coefficients that are in the class of vanishing mean oscillations both in the time and the space variables and that the boundary operators have variable leading coefficients. The proofs are based on and generalize the estimates recently obtained by the authors in [6].

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