scholarly journals Stability and Hölder regularity of solutions to complex Monge–Ampère equations on compact Hermitian manifolds

2021 ◽  
pp. 1-27
Author(s):  
Chinh H. Lu ◽  
Trong-Thuc Phung ◽  
Tât-Dat Tô
Keyword(s):  
Regularity Of Solutions ◽  
Hölder Regularity ◽  
Hermitian Manifolds ◽  
Monge Ampere ◽  
Holder Regularity

scholarly journals Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Nam Q. Le
Keyword(s):  
Boundary Data ◽  
Boundary Regularity ◽  
Hölder Regularity ◽  
Convex Domains ◽  
Open Convex ◽  
Optimal Boundary ◽  
Monge Ampere ◽  
Holder Regularity ◽  
Bounded Convex

<p style='text-indent:20px;'>By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as <inline-formula><tex-math id="M1">\begin{document}$ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $\end{document}</tex-math></inline-formula> with zero boundary data, have unexpected degenerate nature.</p>

Regularity theory of Kolmogorov operator revisited

2020 ◽  
pp. 1-12
Author(s):  
Damir Kinzebulatov
Keyword(s):  
Elliptic Equation ◽  
Vector Fields ◽  
Regularity Theory ◽  
Regularity Of Solutions ◽  
Hölder Regularity ◽  
Kolmogorov Operator ◽  
Holder Regularity ◽  
Critical Order

Abstract We consider Kolmorogov operator $-\Delta +b \cdot \nabla $ with drift b in the class of form-bounded vector fields (containing vector fields having critical-order singularities). We characterize quantitative dependence of the Sobolev and Hölder regularity of solutions to the corresponding elliptic equation on the value of the form-bound of b.

scholarly journals New interpolation spaces and strict Hölder regularity for fractional abstract Cauchy problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Md Mansur Alam ◽  
Shruti Dubey ◽  
Dumitru Baleanu
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Analytic Semigroup ◽  
Mild Solutions ◽  
Abstract Cauchy Problem ◽  
Regularity Of Solutions ◽  
Hölder Regularity ◽  
Interpolation Spaces ◽  
Holder Regularity

AbstractWe know that interpolation spaces in terms of analytic semigroup have a significant role into the study of strict Hölder regularity of solutions of classical abstract Cauchy problem (ACP). In this paper, we first construct interpolation spaces in terms of solution operators in fractional calculus and characterize these spaces. Then we establish strict Hölder regularity of mild solutions of fractional order ACP.

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