Maximum principles, gradient estimates, and weak solutions for second-order partial differential equations

The Maximum Principle for Viscosity Solutions of Fully Nonlinear Second Order Partial Differential Equations

We prove that viscosity solutions in W 1'~176 of the second order, fully nonlinear, equation F(D2u, Du, u) = 0 are unique when (i) F is degenerate elliptic and decreasing in u or (ii) Fis uniformly elliptic and nonincreasing in u. We do not assume that F is convex. The method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions ont...

global results on some nonlinear partial differential equations for direct and inverse problems

در این رساله به بررسی رفتار جواب های رده ای از معادلات دیفرانسیل با مشتقات جزیی در دامنه های کراندار می پردازیم . این معادلات به فرم نیم-خطی و غیر خطی برای مسایل مستقیم و معکوس مورد مطالعه قرار می گیرند . به ویژه، تاثیر شرایط مختلف فیزیکی را در مساله، نظیر وجود موانع و منابع، پراکندگی و چسبندگی در معادلات موج و گرما بررسی می کنیم و به دنبال شرایطی می گردیم که متضمن وجود سراسری یا عدم وجود سراسر...

On Approximate Solutions of Second-Order Linear Partial Differential Equations

In this paper, a Chebyshev polynomial approximation for the solution of second-order partial differential equations with two variables and variable coefficients is given. Also, Chebyshev matrix is introduced. This method is based on taking the truncated Chebyshev expansions of the functions in the partial differential equations. Hence, the result matrix equation can be solved and approximate va...

Gradient Estimates for Weak Solutions of A-Harmonic Equations

Strong and weak error estimates for the solutions of elliptic partial differential equations with random coefficients

We consider the problem of numerically approximating the solution of an elliptic partial di erential equation with random coe cients and homogeneous Dirichlet boundary conditions. We focus on the case of a lognormal coe cient, we have then to deal with the lack of uniform coercivity and uniform boundedness with respect to the randomness. This model is frequently used in hydrogeology. We approxi...