Differential and Integral Equations

A priori estimates for elliptic problems with a strongly singular gradient term and a general datum

Daniela Giachetti, Francesco Petitta, and Sergio Segura de León

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Abstract

In this paper we show approximation procedures for studying singular elliptic problems whose model is $$ \begin{cases} - \Delta u= b(u)|\nabla u|^2+f(x), & \text{in } \; \Omega;\\ u = 0, & \text{on } \; \partial \Omega; \end{cases} $$ where $b(u)$ is singular in the $u$-variable at $u=0$, and $f \in L^m (\Omega)$, with $m>\frac N2$, is a function that does not have a constant sign. We will give an overview of the landscape that occurs when different problems (classified according to the sign of $b(s)$) are considered. So, in each case and using different methods, we will obtain a priori estimates, prove the convergence of the approximate solutions, and show some regularity properties of the limit.

Article information

Source
Differential Integral Equations, Volume 26, Number 9/10 (2013), 913-948.

Dates
First available in Project Euclid: 3 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1372858556

Mathematical Reviews number (MathSciNet)
MR3100071

Zentralblatt MATH identifier
1299.35139

Subjects
Primary: 35J75, 35J25: Boundary value problems for second-order elliptic equations

Citation

Giachetti, Daniela; Petitta, Francesco; de León, Sergio Segura. A priori estimates for elliptic problems with a strongly singular gradient term and a general datum. Differential Integral Equations 26 (2013), no. 9/10, 913--948. https://projecteuclid.org/euclid.die/1372858556


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