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Article Dans Une Revue Selecta Mathematica Année : 2004

Singularities of positive supersolutions in elliptic PDEs

Résumé

Let $\\Omega\\subset\\Bbb{R}^N$ be a bounded domain and denote by ${\\rm cap}_2$ the standard $H^1$-capacity. For any Radon measure $µ$ in $\\Bbb{R}^N$, consider the \"Radon-Nikodym\" decomposition $µ=\\mu_{\\rm d}+\\mu_{\\rm c}$ with respect to ${\\rm cap}_2$, so that the diffuse measure $\\mu_{\\rm d}$ satisfies $\\mu_{\\rm d}(A)=0$ for any Borel set $A\\subset\\Omega$ with ${\\rm cap}_2(A)=0$, and $|\\mu_{\\rm c}|(\\Omega\\sbs F)=0$ for some Borel set $F\\subset\\Omega$ such that ${\\rm cap}_2(F)=0$. In the paper under review the authors discuss the question: \"When can the set of singularities of a solution to a linear (or quasi-linear) elliptic equation be removed?\" The authors prove the following main results: Theorem 1. Assume that ${\\rm cap}_2(\\Sigma)=0$. Let $c\\in\\Bbb{R}$ and $f\\in L^1_{\\rm loc}(\\Omega)$. If $u\\in L^1_{\\rm loc}(\\Omega\\sbs\\Sigma)$, $u\\geq0$ a.e. in $\\Omega$, satisfies $-\\Delta u+cu\\geq f$ in $\\scr{D}\'(\\Omega\\sbs\\Sigma)$, then $u\\in L^1_{\\rm loc}(\\Omega)$, and $-\\Delta u+cu\\geq f$ in $\\scr{D}\'(\\Omega)$. Theorem 3 (Ïnverse maximum principle\"). Let $u\\in L^1_{\\rm loc}(\\Omega)$ be such that $\\Delta u$ is a Radon measure in $\\Omega$. If $u\\geq0$ a.e. in $\\Omega$, then $(-\\Delta u)_{\\rm c}\\geq0$. Extensions to general second-order linear and quasi-linear equations are also discussed.
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Dates et versions

hal-00009121 , version 1 (27-09-2005)

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  • HAL Id : hal-00009121 , version 1

Citer

Louis Dupaigne, Augusto C. Ponce. Singularities of positive supersolutions in elliptic PDEs. Selecta Mathematica, 2004, 10, pp.341--358. ⟨hal-00009121⟩
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