scholarly journals Maximal regularity for abstract parabolic problems with inhomogeneous boundary data in $L_p$-spaces

2002 ◽  
Vol 127 (2) ◽  
pp. 311-327 ◽  
Author(s):  
Jan Prüss
Keyword(s):  
Boundary Data ◽  
Maximal Regularity ◽  
Parabolic Problems ◽  
Inhomogeneous Boundary Data

scholarly journals Inverse parabolic problems of determining functions with one spatial-component independence by Carleman estimate

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Oleg Y. Imanuvilov ◽  
Yavar Kian ◽  
Masahiro Yamamoto
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Boundary Data ◽  
Stability Estimate ◽  
Carleman Estimate ◽  
Parabolic Problems ◽  
Conditional Stability ◽  
Inverse Source Problem ◽  
Lateral Boundary ◽  
Spatial Component

Abstract For a parabolic equation in the spatial variable x = ( x 1 , … , x n ) {x=(x_{1},\ldots,x_{n})} and time t, we consider an inverse problem of determining a coefficient which is independent of one spatial component x n {x_{n}} by lateral boundary data. We apply a Carleman estimate to prove a conditional stability estimate for the inverse problem. Also, we prove similar results for the corresponding inverse source problem.

scholarly journals Maximal regularity for elliptic operators with second-order discontinuous coefficients

2020 ◽  
Author(s):  
G. Metafune ◽  
L. Negro ◽  
C. Spina
Keyword(s):  
Elliptic Operator ◽  
Maximal Regularity ◽  
Discontinuous Coefficients ◽  
Elliptic Operators ◽  
Second Order ◽  
Parabolic Problems ◽  
Order Elliptic Operator

Abstract We prove maximal regularity for parabolic problems associated to the second-order elliptic operator $$\begin{aligned} L =\Delta +(a-1)\sum _{i,j=1}^N\frac{x_ix_j}{|x|^2}D_{ij}+c\frac{x}{|x|^2}\cdot \nabla -b|x|^{-2} \end{aligned}$$ L = Δ + ( a - 1 ) ∑ i , j = 1 N x i x j | x | 2 D ij + c x | x | 2 · ∇ - b | x | - 2 with $$a>0$$ a > 0 and $$b,\ c$$ b , c real coefficients.

scholarly journals Mixed problems for degenerate abstract parabolic equations and applications

2018 ◽  
Vol 34 (2) ◽  
pp. 247-254
Author(s):  
VELI. B. SHAKHMUROV ◽  
◽  
AIDA SAHMUROVA ◽  
◽  
Keyword(s):  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Maximal Regularity ◽  
Variable Coefficients ◽  
Parabolic Problems ◽  
Mixed Problems ◽  
Regularity Properties ◽  
Nonlinear Parabolic ◽  
Abstract Parabolic Equations ◽  
Mixed Lebesgue Spaces

Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.

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