Semi-linear second-order elliptic equations in L1
Haïm Brézis, Walter A. Strauss · Journal of the Mathematical Society of Japan · 1973
a paper which is yet to be completed, we have obtained some analogous results for parabolic equations.\S 1.An abstract formulation of the monotone case.Let $\beta$ be a maximal monotone graph in $R\times R$ which contains the origin.If the pair $(s, t)\in\beta$ , we write $t\in\beta(s)$ .Let $\Omega$ be any measure space.We denote by $\Vert\Vert_{p}$ the norm in $L^{p}(\Omega)$ .Let $A$ be an unbounded linear operator on $L^{1}(\Omega)$ which satisfies the following conditions. (I) It is a (closed) operator with dense domain $D(A)$ in $L^{1}(\Omega)$ ; for any(II) For any $\lambda>0$ and $f\in L^{1}(\Omega)$ , $\sup_{\Omega}(I+\lambda A)^{-1}f\leqq\max\{0, \sup_{\Omega}f\}$ .(By " sup" we mean the essential supremum.If $supf=\infty$ , assumption (II) is empty.) (III) There exists $\alpha>0$ such that $\alpha\Vert u\Vert_{1}\leqq\Vert$ Au $\Vert_{1}$ for all $u\in D(A)$ .THEOREM 1.For every $f\in L^{1}(\Omega)$ , there exists a unique $u\in D(A)$ such that (2) Au $(x)+\beta(u(x)) i f(x)$ $a$ .$e$ .Moreover, if $f,$ $f\in L^{1}(\Omega)$ and $u,$ \^u are the correspOnding solutions of (2), then(3)In particular, (4)LEMMA 2. Let $\gamma$ be a maximal monotone graph in $R\times R$ which contains the origin.Assume that $A$ satisfies (I) and (II).Let $ 1\leqq P\leqq\infty$ and $P^{\prime}=p/(p-1)$ , $ p^{\prime}=\infty$ if $p=1$ .Let $u\in D(A)\cap L^{p}(\Omega)$ with $Au\in L^{p}(\Omega)$ .Let $g\in L^{p\prime}(\Omega)$ be such that $g(x)\in\gamma(u(x))a$ .$e$ .Then $\int_{\Omega}Au(x)g(x)dx\geqq 0$ .PROOF OF THEOREM 1.We denote, for $u$ and $f\in L^{1}(\Omega),$ $f\in Bu$ whenever $f(x)\in\beta(u(x))a$ .$e$ .We first establish (3) which implies (4) and the uniqueness.Let $g=f-Au\in Bu$ and $\hat{g}=\hat{f}-A\hat{u}\in B\text{{\it \^{u}}}$ .We multiply the equation $*(I)$ is equivalent to $-A$ generating a linear contraction semi.group in $L^{1}(\Omega)$ .