{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:THU2THKMWOIPU3C23JVSLSGJWE","short_pith_number":"pith:THU2THKM","schema_version":"1.0","canonical_sha256":"99e9a99d4cb390fa6c5ada6b25c8c9b121250c4534dba9013b46bf4c0df78c87","source":{"kind":"arxiv","id":"1604.00520","version":1},"attestation_state":"computed","paper":{"title":"A natural approach to the asymptotic mean value property for the $p$-Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hidemitsu Wadade, Michinori Ishiwata, Rolando Magnanini","submitted_at":"2016-04-02T16:09:02Z","abstract_excerpt":"Let $1\\le p\\le\\infty$. We show that a function $u\\in C(\\mathbb R^N)$ is a viscosity solution to the normalized $p$-Laplace equation $\\Delta_p^n u(x)=0$ if and only if the asymptotic formula $$ u(x)=\\mu_p(\\ve,u)(x)+o(\\ve^2) $$ holds as $\\ve\\to 0$ in the viscosity sense. Here, $\\mu_p(\\ve,u)(x)$ is the $p$-mean value of $u$ on $B_\\ve(x)$ characterized as a unique minimizer of $$ \\inf_{\\la\\in\\RR}\\nr u-\\la\\nr_{L^p(B_\\ve(x))}. $$ This kind of asymptotic mean value property (AMVP) extends to the case $p=1$ previous (AMVP)'s obtained when $\\mu_p(\\ve,u)(x)$ is replaced by other kinds of mean values. 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We show that a function $u\\in C(\\mathbb R^N)$ is a viscosity solution to the normalized $p$-Laplace equation $\\Delta_p^n u(x)=0$ if and only if the asymptotic formula $$ u(x)=\\mu_p(\\ve,u)(x)+o(\\ve^2) $$ holds as $\\ve\\to 0$ in the viscosity sense. Here, $\\mu_p(\\ve,u)(x)$ is the $p$-mean value of $u$ on $B_\\ve(x)$ characterized as a unique minimizer of $$ \\inf_{\\la\\in\\RR}\\nr u-\\la\\nr_{L^p(B_\\ve(x))}. $$ This kind of asymptotic mean value property (AMVP) extends to the case $p=1$ previous (AMVP)'s obtained when $\\mu_p(\\ve,u)(x)$ is replaced by other kinds of mean values. 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