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The Reverse Poincare Inequalities for Weak Subsolutions of Elliptic and Parabolic Equations

Thesis Info

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Author

Saleem, Muhammad Shoaib

Program

PhD

Institute

Government College University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2006

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/handle/123456789/1690

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727593930

Similar


The continuous weak subsolutions of general type second order linear partial dif- ferential equations are studied in the present thesis. Based on monotonic approximation techniques developed by Walter Littman (1963) we prove that under some regularity conditions on the coefficients of the uniformly elliptic differential operator any bounded continuous weak subsolution in a smooth domain D possesses all first order weak (Sobolev) partial derivatives and belongs to the weighted Sobolev space H 1 (D; h), where h(x) is the appropriate weight function. Moreover, we establish a new type weighted reverse Poincare inequality for the dif- ference of two bounded and continuous weak subsolutions. Further the latter inequality is applied to the approximation problem of the gradient of the analytically unknown value function of the optimal stochastic control prob- lem, the value function being the unique solution of the Hamilton-Jacobi-Bellman equation.
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