Heriot-Watt Mathematics Report Series
HWM01-6, 2 March 2001
Strong versus weak local minimizers for the perturbed Dirichlet functional
A Taheri
Abstract
We consider integral functionals of the from
$$
I(u)= \int_{\Omega} \left( \frac{1}{2} |Du|^2 + F(x,u) \right) \, dx
$$
where $F: \Omega \times R}^N \to R}$ satsifies the growth $F(x,u) \ge -C (1+|u|^p)$ for some $C>0$ and $1 \le p < \infty$. We state and prove a sufficiency theorem for $L^r$ local minimizers of $I$ where $1 \le r \le \infty$. The exponent $r$ is shown to depend on the dimension $n$ and $p$ and an exact expression is presented for this dependence.
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