scholarly journals Hölder Regularity of the 2D Dual Semigeostrophic Equations via Analysis of Linearized Monge–Ampère Equations

2018 ◽  
Vol 360 (1) ◽  
pp. 271-305 ◽  
Author(s):  
Nam Q. Le
Keyword(s):  
Hölder Regularity ◽  
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>

scholarly journals Hölder regularity for degenerate parabolic equations with variable exponents

2022 ◽  
Vol 40 ◽  
pp. 1-19
Author(s):  
Hamid EL Bahja
Keyword(s):  
Weak Solutions ◽  
Parabolic Equations ◽  
Degenerate Parabolic Equations ◽  
Hölder Regularity ◽  
Variable Exponents ◽  
Young’S Inequality ◽  
Scaling Method ◽  
Degenerate Parabolic ◽  
Young's Inequality ◽  
Holder Regularity

In this paper, we discuss a class of degenerate parabolic equations with variable exponents. By  using the Steklov average and Young's inequality, we establish energy and logarithmicestimates for solutions to these equations. Then based on the intrinsic scaling method, we provethat local weak solutions are locally continuous.

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