Abstract
AbstractFirst we prove a comparison result for a nonlinear divergence structure elliptic partial differential equation. Next we find an estimate of the solution of a boundary value problem in a domainΩin terms of the solution of a related symmetric boundary value problem in a ballBhaving the same measure asΩ. For p-Laplace equations, the corresponding result is due to Giorgio Talenti. In a special (radial) case we also prove a reverse comparison result.
Highlights
In the seminal paper [1], Giorgio Talenti established sharp estimates of the solution to a boundary value problem of a second order elliptic partial di erential equation in terms of the solution of a related symmetric problem
First we prove a comparison result for a nonlinear divergence structure elliptic partial di erential equation
For pLaplace equations, the corresponding result is due to Giorgio Talenti
Summary
In the seminal paper [1], Giorgio Talenti established sharp estimates of the solution to a boundary value problem of a second order elliptic partial di erential equation in terms of the solution of a related symmetric problem. We refer to the survey [2] for a detailed treatment of the subject. The interest of these results relies on the obvious fact that a symmetric problem reduces to an ordinary di erential equation and is easier to be solved. Let g be positive and such that g(s )s is strictly increasing and di erentiable for s >. If B ⊂ Rn is the ball centered at the origin with the same measure as Ω and if f is the Schwarz (decreasing) rearrangement of f , let v be a solution to. Under suitable conditions on the function h, the answer is positive for the p-Laplacian, where g(s ) = sp− , p >. Many authors have studied (p, q)-Laplace equations, see [4,5,6] and references
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