April  2002, 8(2): 319-329. doi: 10.3934/dcds.2002.8.319

Some Dirichlet problems with bad coercivity

1. 

Dipartimento di Matematica, Università di Roma 1, Piazza A. Moro 2, 00185 Roma, Italy

Revised  October 2001 Published  January 2002

Here I summarize and I translate in English my lectures, devoted to some Dirichlet problems with a common feature: bad coercivity.
Very simple examples are:

$-\Delta u = f(x)\in L^1(\Omega)\quad$ in $\Omega$

$u = 0\quad$ on $\partial \Omega$

since the term $\int_\Omega f(x)v(x)$ does not make sense, if $f\in L^1(\Omega), v\in W^{1,2}_0(\Omega)$;

-div$(\frac{\nabla u}{(1+|u|)^\theta})=f(x)\in L^2(\Omega)\quad$ in $\Omega$

$u = 0\quad$ on $\partial \Omega$

Since the term $\int_\Omega \frac{|\nabla v|^2}{(1+|v|)^\theta}$ goes to zero, if $v$ is large.

Citation: Lucio Boccardo. Some Dirichlet problems with bad coercivity. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 319-329. doi: 10.3934/dcds.2002.8.319
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