On Probabilistic Analytical and Numerical Approaches for Divergence Form Operators with Discontinuous Coefficients

2014 ◽  
pp. 267-296
Author(s):  
Denis Talay
Keyword(s):  
Discontinuous Coefficients ◽  
Divergence Form ◽  
Numerical Approaches

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<∞.

scholarly journals AnLp-Estimate for Weak Solutions of Elliptic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Sara Monsurrò ◽  
Maria Transirico
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
A Priori ◽  
Unbounded Domains ◽  
Discontinuous Coefficients ◽  
Elliptic Partial Differential Equations ◽  
Divergence Form ◽  
A Priori Bound

We prove anLp-a priori bound,p>2, for solutions of second order linear elliptic partial differential equations in divergence form with discontinuous coefficients in unbounded domains.

scholarly journals Explicit Estimates for Solutions of Mixed Elliptic Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Luisa Consiglieri
Keyword(s):  
Potential Theory ◽  
Mixed Problem ◽  
Elliptic Problems ◽  
Discontinuous Coefficients ◽  
Divergence Form ◽  
Order Equation ◽  
Second Order ◽  
Green Functions ◽  
Mixed Problems ◽  
Quantitative Estimates

We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second-order equation in divergence form with discontinuous coefficients. Our concern is to estimate the solutions with explicit constants, for domains in ℝn (n≥2) of class C0,1. The existence of L∞ and W1,q estimates is assured for q=2 and any q<n/(n-1) (depending on the data), whenever the coefficient is only measurable and bounded. The proof method of the quantitative L∞ estimates is based on the De Giorgi technique developed by Stampacchia. By using the potential theory, we derive W1,p estimates for different ranges of the exponent p depending on the fact that the coefficient is either Dini-continuous or only measurable and bounded. In this process, we establish new existences of Green functions on such domains. The last but not least concern is to unify (whenever possible) the proofs of the estimates to the extreme Dirichlet and Neumann cases of the mixed problem.

scholarly journals A Priori Bounds in and in for Solutions of Elliptic Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Sara Monsurrò ◽  
Maria Transirico
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equations ◽  
A Priori ◽  
Unbounded Domains ◽  
Variational Problems ◽  
Discontinuous Coefficients ◽  
Elliptic Partial Differential Equations ◽  
Divergence Form ◽  
A Priori Bounds ◽  
A Priori Bound

We give an overview on some recent results concerning the study of the Dirichlet problem for second-order linear elliptic partial differential equations in divergence form and with discontinuous coefficients, in unbounded domains. The main theorem consists in an -a priori bound, . Some applications of this bound in the framework of non-variational problems, in a weighted and a non-weighted case, are also given.

scholarly journals A Property of Weak Solutions for Some Parabolic Equations of Higher Order

1967 ◽  
Vol 29 ◽  
pp. 57-60
Author(s):  
Lu-San Chen
Keyword(s):  
Weak Solutions ◽  
Parabolic Equations ◽  
Analogous Result ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Discontinuous Coefficients ◽  
Divergence Form ◽  
Higher Order ◽  
Uniqueness Property ◽  
Initial Boundary Value

Recently Aronson [1] proved the uniqueness property of weak solutions of the initial boundary value problem for second order parabolic equations with discontinuous coefficients. An analogous result to Aronson’s was proved by Kuroda M in the case of some parabolic equations of higher order, where the method due to Aronson [2] plays an essential role. In this paper, under the same idea we shall be concerned with the asymptotic behavior of weak solutions for parabolic equations of higher order of the divergence form, when the data are prescribed on a portion of a time-like surface.

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