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A fourth order nonlinear degenerate parabolic equation
Regularity for positive weak solutions to semilinear elliptic equations
1.  Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 
$\Delta u= f(u)\quad$ in $\Omega,$
with $f(u)\in L^1(\Omega)$. Here $\Omega$ is a bounded domain in $R^n$, and $f(u)$ is a regular function with respect to $u$. We give an apriori estimate for positive weak solutions. We show that under some appropriate assumptions on the nonlinear term $f$, the positive weak solutions are in fact in some local Sobolev space $W_{l o c}^{1,\tau}(\Omega)$. We also derive a very general local monotonicity formula for variational solutions to the equation above with special nonlinear term $f$.
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