scholarly journals The Dirichlet problem for second order semilinear elliptic and parabolic equations

2009 ◽  
pp. 393-411
Author(s):  
Alkis Tersenov
Keyword(s):  
Dirichlet Problem ◽  
Parabolic Equations ◽  
Second Order ◽  
Semilinear Elliptic ◽  
Elliptic And Parabolic Equations

Perturbation and Solvability of Initial Lp Dirichlet Problems for Parabolic Equations over Non-cylindrical Domains

2014 ◽  
Vol 66 (2) ◽  
pp. 429-452 ◽  
Author(s):  
Jorge Rivera-Noriega
Keyword(s):  
Dirichlet Problem ◽  
Parabolic Equations ◽  
Elliptic Equations ◽  
Carleson Measure ◽  
Divergence Form ◽  
Second Order ◽  
Linear Operators ◽  
Dirichlet Problems ◽  
Small Perturbations ◽  
Cylindrical Domains

AbstractFor parabolic linear operators L of second order in divergence form, we prove that the solvability of initial Lp Dirichlet problems for the whole range 1 < p < ∞ is preserved under appropriate small perturbations of the coefficients of the operators involved. We also prove that if the coefficients of L satisfy a suitable controlled oscillation in the form of Carleson measure conditions, then for certain values of p > 1, the initial Lp Dirichlet problem associated with Lu = 0 over non-cylindrical domains is solvable. The results are adequate adaptations of the corresponding results for elliptic equations.

scholarly journals Partial Schauder Estimates for Second-Order Elliptic and Parabolic Equations: A Revisit

2017 ◽  
Vol 2019 (7) ◽  
pp. 2085-2136 ◽  
Author(s):  
Hongjie Dong ◽  
Seick Kim
Keyword(s):  
Parabolic Equations ◽  
Divergence Form ◽  
Second Order ◽  
The Other ◽  
Schauder Estimates ◽  
Independent Variables ◽  
Elliptic And Parabolic Equations

AbstractUnder various conditions, we establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving Hölder semi-norms not with respect to all, but only with respect to some of the independent variables. A novelty of our results is that the coefficients are allowed to be merely measurable with respect to the other independent variables.

CAUCHY–DIRICHLET PROBLEM ASSOCIATED TO DIVERGENCE FORM PARABOLIC EQUATIONS

2004 ◽  
Vol 06 (03) ◽  
pp. 377-393 ◽  
Author(s):  
MARIA ALESSANDRA RAGUSA
Keyword(s):  
Parabolic Equation ◽  
Dirichlet Problem ◽  
Parabolic Equations ◽  
Discontinuous Coefficients ◽  
Divergence Form ◽  
Second Order ◽  
Linear Parabolic Equation

In this note we study the Cauchy–Dirichlet problem related to a linear parabolic equation of second order in divergence form with discontinuous coefficients. Moreover we prove estimates in the space [Formula: see text], for every 1<p<∞.

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